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This article is cited in 2 scientific papers (total in 2 papers) Iterates of holomorphic maps, fixed points, and domains of univalence V. V. Goryainova, O. S. Kudryavtsevabc, A. P. Solodovb a Moscow Institute of Physics and Technology (National Research University) b Lomonosov Moscow State University, Moscow Center for Fundamental and Applied Mathematics c Volgograd State Technical University
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     1. IntroductionThe problem of fractional iterations of a holomorphic map has a rich history. Many authors have addressed this problem in various settings. Let $f$ be a holomorphic function such that its values lie in its domain of definition. Then its iterates $f^0(z)=z$, $f^{1}(z)=f(z)$, and $f^{n}(z)=f\circ f^{n-1}(z)$, $n=2,3,\dots$, are well defined. A natural question is whether or not there exists a family of functions $f^t(z)$, $t\geqslant 0$, such that $f^1(z)=f(z)$, $f^0(z)=z$, and $f^{t+s}(z)=f^t\circ f^s(z)$ for all $s,t \geqslant 0$. For non-negative integers $t$ we obtain the ordinary iterates of $f=f^1$. So some authors call elements of the family $\{f^t\}_{t\geqslant 0}$ fractional iterates of $f$. The first investigations of the problem of fractional iterations concerned the local case, when both $f$ and its iterates are only defined by power series in a neighbourhood of a fixed point $a$, $f(a)=a$. As long ago as the 1880s, Koenigs showed that if $|f'(a)|$ is distinct from $0$ and $1$, then $f$ can be embedded in a family of fractional iterates. Close links between the problem of fractional iterations and solutions to Schröder’s and Abel’s functional equations were also discovered. In the middle of the 20th century the problem of fractional iterations was extensively investigated for entire and meromorphic functions. Results obtained there were quite opposite to the local case. For example, Baker, and also Karlin and McGregor, showed that under quite general assumptions, such an embedding is possible in this case only when $f$ is linear fractional. The links between the problem of fractional iterates and the problem of embedding a Galton-Watson process in a homogeneous branching Markov process with continuous time, as well as some problems in the theory of composition operators and the theory of conformal mappings, were motivations for considering the case when the function and its iterates map the unit disc $\mathbb D=\{z\in \mathbb C\colon |z|<1\}$ (or upper half-plane) into itself. This case is very different from the first two: it is more difficult to investigate and much more prolific in terms of results. As in dynamical systems, fixed points of the map are important in the investigations of iterations. However, a holomorphic self-map of the unit disc need not have fixed points in the interior of the disc. On the other hand, for such a map we can always distinguish the so-called Denjoy–Wolff point, which is the limit of the sequence of (integer) iterates. If it lies in the interior of the disc, then it is fixed. When it occurs on the boundary of the unit disc, it is also fixed in the sense of angular limits. Apart from the Denjoy–Wolff point, a holomorphic self-map of the unit disc can also have other fixed points, but all of these must lie on the unit circle $\mathbb T=\{z\in \mathbb C\colon |z|=1\}$. Thus, an investigation of iterates of a holomorphic self-map of the unit disc is linked with the analysis of its behaviour in a neighbourhood of boundary fixed points. Taking this approach, some classical results related to domains of univalence of holomorphic functions are added new colours. The well-known Bloch constant problem and some related problems deal with discs of univalence and covering discs of holomorphic functions with fixed point and prescribed derivative at this point. Let $f$ be holomorphic in the unit disc $\mathbb D$, $f(0)=0$, and $f'(0)=1$. Then $f$ is univalent in some disc with centre at the origin. However, there exists no disc of fixed radius with centre zero such that all such functions are univalent in it. The situation changes if the additional condition of boundedness is imposed on the function. So let $f$ be holomorphic in the disc $\mathbb D$, fix the origin, and satisfy $f'(0)=1$ and $|f(z)|\leqslant M$, $M>1$, for $z\in \mathbb D$. Then Landau showed that $f$ is univalent in the disc $\{|z|<M-\sqrt{M^2-1}\}$ . This can be reformulated for holomorphic self-maps of the unit disc. Let $\mathscr B$ denote the class of functions $f$ that are holomorphic in the unit disc $\mathbb D$ and map it into itself. If $f\in \mathscr B$ and $f(0)=0$, then by Schwarz’s lemma $|f'(0)|\leqslant 1$, and the equality $|f'(0)|=1$ holds only when $f(z)=\varkappa z$, $|\varkappa|=1$. Now let $f\in \mathscr B$, $f(0)=0$, and $|f'(0)|\geqslant\beta$, $0<\beta <1$. Then all Landau’s assumptions hold for the function $g(z)=f(z)/f'(0)$ and $M=1/\beta$, so this function is univalent in the disc $\{|z| < \beta/(1+\sqrt{1-\beta^2}\,)\}$. The function $f$ itself is also univalent there. It was discovered recently that when $f$ in $\mathscr B$ has two fixed points, then some domain of univalence and some covering domain are also guaranteed under certain assumptions. Such results are the subject of § 6, while in § 4 we present a survey of the classical results on discs of univalence and covering discs. The structure of subsets of the class $\mathscr B$ corresponding to prescribed fixed points is described in § 5. The integral representations from that section are important for the investigation of the properties of holomorphic maps $f$ in $\mathscr B$ with prescribed fixed points. Since all fixed points, apart perhaps from the Denjoy–Wolff point, lie on the boundary, conditions for the existence of domains of univalence and covering domains are naturally stated in terms of angular derivatives. We present important inequalities involving angular derivatives in § 2. In § 3 we give a brief survey of conditions for the univalence of holomorphic functions. Turning back to the problem of fractional iterations we note that the class $\mathscr B$ is a topological semigroup with respect to composition, in the topology of locally uniform convergence. Furthermore, the question of the existence of fractional iterates can be viewed as one relating to the possibility of embedding the integer iterates of $f$ in a one-parameter semigroup. In § 7 we present results describing the form of the infinitesimal generator of a one-parameter semigroup in its dependence on the fixed point set of the map. Section 8 is devoted to the problem of embedding iterates in a one-parameter semigroup. Apart from an embeddability criterion in terms of solutions of Scröder’s and Abel’s functional equations, we state a criterion based on the asymptotic behaviour of the (positive integer) iterates of the function $f$ in $\mathscr B$. Again, the fixed points of $f$ play an important role here. We also obtain integral representations for Koenigs functions, which enable us to recover a one-parameter semigroup of maps with prescribed fixed points. We give special attention to the case of probability generating functions. In particular, we present a solution of the problem of embedding a Galton–Watson process in a homogeneous branching Markov process with continuous time, and we provide simple necessary conditions for such an embedding to exist in terms of the range of values of the initial probabilities. The coefficient problem is a core topic of investigations: suffice it to mention Bieberbach’s conjecture, which has played a significant role in shaping the development of geometric function theory. In the last section, § 9, we investigate the impact of the angular derivatives of a function $f$ in $\mathscr B$ at boundary fixed points on the region of values of its Taylor coefficients. We distinguish two cases: when the coefficients are considered independently of one another, and vice versa. 2. Fixed points and bounds for derivativesAs pointed out in the introduction, fixed points of a function $f\in \mathscr B$ have a decisive influence on its geometric and analytic properties. Among the classical results underlying virtually all research concerned with the properties of functions in $\mathscr B$, there are fundamental inequalities for an interior and a boundary fixed point. It is well known that, apart from the disc $\mathbb D$, a function in $\mathscr B[0]=\{f\in \mathscr B$: $f(0)= 0\}$ also maps every disc with centre $z=0$ into itself. Thus, this is an attracting fixed point, except when we have a rotation of $\mathbb D$. Theorem 1 (Schwarz’s lemma). Let $f\in \mathscr B[0]$. Then and If equality holds in (2.2) or equality holds in (2.1) for at least one point $z\in\mathbb{D}$, $z\ne 0$, then $f(z)=\varkappa z$, where $|\varkappa|=1$. In this case equality in (2.1) holds for all $z\in \mathbb D$. We can treat inequality (2.2) as a solution of the following extremal problem: find the least upper bound in the class $\mathscr B[0]$ for the modulus of the derivative of the function at zero. By Theorem 1, and extremal functions are rotations of the unit disc $\mathbb D$.Pick observed that any map of $\mathbb D$ into itself can be obtained as a composition of a linear fractional automorphism of the disc and a function in $\mathscr B[0]$. This brought him to an invariant version of Schwarz’s lemma, which holds for all functions in $\mathscr B$. Theorem 2 (Schwarz–Pick lemma). Let $f\in \mathscr B$. Then for each $z\in \mathbb D$ We see from (2.3) that the image of any non-Euclidean disc with centre $z_0$ lies in the non-Euclidean disc with centre $f(z_0)$ and the same radius. Inequality (2.4) can be interpreted as a solution of the following extremal problem: find the least upper bound for the modulus of the derivative at $z_0$ in the class of functions in $\mathscr B$ that take a fixed point $z_0\in \mathbb D$ to a prescribed point $w_0$. By Theorem 2 we have where the supremum is taken over all $f\in\mathscr B$ satisfying $f(z_0)=w_0$. In this case extremal functions are linear fractional self-maps of the unit disc $\mathbb D$ taking $z_0$ to $w_0$.Dieudonné [1] showed that in $\mathscr B[0]$ the modulus of the derivative is bounded by 1 not only at the point $z=0$ (see (2.2)), but also in the whole of the disc $\{z\in \mathbb D\colon |z|\leqslant\sqrt{2}-1\}$. This result is based on the following estimate for the derivative in the class $\mathscr B[0]$. Theorem 3 (Dieudonné [1]). Let $f\in \mathscr B[0]$. Then for each $z\in \mathbb D$, $z\ne 0$, To prove Theorem 3 it suffices to apply (2.4) to $f(z)/z$. By Theorem 3, on the class of functions in $\mathscr B[0]$ taking a fixed point $z_0\in \mathbb D$ to a prescribed point $w_0$ the set of values $f'(z_0)$ of the derivative belongs to the disc
$$
\begin{equation*}
\biggl|f'(z_0)-\frac{w_0}{z_0}\biggr|\leqslant \frac{|z_0|^2-|w_0|^2}{|z_0|(1-|z_0|^2)}\,.
\end{equation*}
\notag
$$
Moreover, $f'(z_0)$ lies on the boundary of this disc if and only if $f$ is a second-order Blaschke product such that $f(0)=0$ and $f(z_0)=w_0$. Now we show how for functions $f$ in $\mathscr B[0]$ Theorem 3 yields the estimate $|f'(z)|\leqslant 1$ for $|z|\leqslant \sqrt{2}-1$. Fix $z\in\mathbb D$ and set $t=|f(z)|$. Then by Theorem 3
$$
\begin{equation*}
|f'(z)|\leqslant \frac{t}{|z|}+\frac{|z|^2-t^2}{|z|(1-|z|^2)}= \frac{(1-t)(|z|^2+t)}{|z|(1-|z|^2)}\,.
\end{equation*}
\notag
$$
By Schwarz’s lemma $t\leqslant |z|$, and if $|z|\leqslant \sqrt{2}-1$, then the quadratic function $(1-t)(|z|^2+t)$ is increasing and attains its maximum at $t=|z|$. Thus, $|f'(z)|\leqslant 1$. One consequence of the Schwarz–Pick lemma is as follows: a function $f\in \mathscr B$, $f(z)\not\equiv z$, can have at most one fixed point in the interior of ${\mathbb D}$. We say that a boundary point $a$ is a fixed point of $f$ if $\angle\lim_{z\to a}f(z)=a$. It is known (for instance, see [2]) that the (finite or infinite) angular limit exists at a boundary fixed point. Moreover, if it is finite, then it is a positive number and $f'(z)$ has the same angular limit as $z\to a$. In this case we call (2.6) the angular derivative of $f$ at the point $z=a$ and denote it by $f'(a)$.The classical lower bound for the angular derivative at a boundary fixed point was established by Julia and, independently, by Carathéodory. Let $\mathscr B\{1\}$ be the class of functions with fixed point $z=1$:
$$
\begin{equation*}
\mathscr B\{1\}=\Bigl\{f\in \mathscr B\colon \angle \lim_{z\to 1} f(z)=f(1)=1\Bigr\}.
\end{equation*}
\notag
$$
Set
$$
\begin{equation*}
L_q(z)=\frac{1-\overline{q}}{1-q}\,\frac{z-q}{1-\overline{q}z}\,, \qquad q\in \mathbb D,
\end{equation*}
\notag
$$
which is a linear fractional map taking $\mathbb {D}$ to itself and satisfying $L_q(q)=0$ and $L_q(1)=1$. By the horocycle at $1$ with parameter $k>0$ we mean the disc
$$
\begin{equation*}
\mathbb H_k=\biggl\{z\in\mathbb D\colon\frac{|1-z|^2}{1-|z|^2}< k\biggr\}
\end{equation*}
\notag
$$
with centre $z_0=1/(k+1)$ and radius $k/(k+1)$, which is tangent to the unit circle $\mathbb T$ at $z=1$. Theorem 4 (Julia–Carathéodory). Let $f$ be a function in $\mathscr B\{1\}$. Then for each $z\in \mathbb D$ In fact, inequality (2.7) means that for each $k>0$ the image of the horocycle $\mathbb H_k$ lies in the horocycle $\mathbb H_{f'(1)k}$. We can view the Julia–Carathéodory theorem as a solution of an extremal problem: find the greatest lower bound for the values $f'(1)$ of the angular derivative on the class of functions in $\mathscr B$ that take a fixed point $z_0\in \mathbb D$ to a prescribed point $w_0$ and fix $z=1$. By Theorem 4 we have
$$
\begin{equation}
\inf f'(1)=\frac{|1-w_0|^2}{1-|w_0|^2}\, \frac{1-|z_0|^2}{|1-z_0|^2}\,,
\end{equation}
\tag{2.8}
$$
where the infimum is taken over all $f\in\mathscr B\{1\}$ such that $f(z_0)=w_0$. The linear fractional self-map of the unit disc satisfying $f(z_0)=w_0$ and $f(1)=1$ is an extremal function here. In particular, in the important class $\mathscr B[0,1]=\mathscr B[0]\cap\mathscr B\{1\}$ of holomorphic self-maps of $\mathbb D$ fixing an interior and a boundary point relation (2.8) assumes the following form: and equality in (2.9) is only attained for $f(z)\equiv z$. The bound (2.9) means that for functions in the class $\mathscr B[0,1]$ the boundary fixed point $z=1$ is repelling.Similarly to Schwarz’s lemma, the Julia–Carathéodory theorem has many consequences. We dwell on some of these. Unkelbach refined (2.9) in the case when the modulus of the derivative at the interior fixed point $z=0$ is known. Theorem 5 (Unkelbach [3]). Let $f\in \mathscr B[0,1]$. Then Moreover, equality holds in (2.10) if and only if Inequality (2.10) is connected with the following extremal problem: find the greatest lower bound for the angular derivative $f'(1)$ on the class of functions in $\mathscr B[0,1]$ with fixed value of $|f'(0)|$. By Theorem 5 we have where the infimum is taken over all $f\in\mathscr B[0,1]$ such that $|f'(0)|=M$ and $M\in [0,1)$. The extremal function has the form $f(z)=z(z+M)/(1+M z)$.Osserman [4] extended Theorem 5 to the situation when the origin is not a fixed point. Theorem 6 (Osserman [4]). Let $f\in \mathscr B\{1\}$. Then In the general case the bound (2.11) cannot be attained; it is sharp only for $f(0) > 0$. We present below a refinement of (2.11) obtained in [5], which cannot be improved. Our arguments will be based on the following result, which describes the set of values $f'(0)$ on the class $\mathscr B[0,1]$ in its dependence on the angular derivative $f'(1)$. Theorem 7 (Goryainov [6]). Let $f\in \mathscr B[0,1]$, $f(z)\not\equiv z$. Then Proof. Since $f$ satisfies the assumptions of Schwarz’s lemma, the function $g(z)=f(z)/z$ maps the unit disc $\mathbb D$ to itself, and $g(1)=1$ because $f(1)=1$. Thus, $g\in\mathscr B\{1\}$. Applyig the Julia–Carathéodory theorem to $g$, for $z\in \mathbb D$ we obtain
$$
\begin{equation*}
\frac{|1-g(z)|^2}{1-|g(z)|^2}\, \frac{1-|z|^2}{|1-z|^2}\leqslant g'(1),
\end{equation*}
\notag
$$
which is equivalent to the inequality
$$
\begin{equation}
\frac{|1-f(z)/z|^2}{1-|f(z)/z|^2}\, \frac{1-|z|^2}{|1-z|^2}\leqslant f'(1)-1.
\end{equation}
\tag{2.13}
$$
Setting $z=0$ in (2.13) we arrive at the bound which is equivalent to (2.12).
We turn to the proof of the second part of the statement. Assume that equality holds in (2.12) for some $f\in \mathscr B[0,1]$, $f(z)\not\equiv z$, or, which is the same, assume that We can write (2.14) in terms of the function $g(z)=f(z)/z$, which belongs to the class $\mathscr B\{1\}$: Now we use the Julia–Carathéodoty theorem, which says that such an equality is only possible for the linear fractional maps $g(z)=L_q(z)$, where $q\in\mathbb D$. Thus, $f(z)=zL_q(z)$, where $q\in\mathbb D$.Conversely, let $f (z)=zL_q(z)$ for some $q\in\mathbb D$. It is obvious that $f\in \mathscr B[0,1]$, and since
$$
\begin{equation*}
f'(0)=-q\,\frac{1-\overline{q}}{1-q}\,, \qquad f'(1)=\frac{2-q-\overline{q}}{(1-q)(1-\overline{q})}\,,
\end{equation*}
\notag
$$
it is easy to see that $f$ makes of (2.12) an equality. $\Box$ To improve Osserman’s inequality (2.11) we require the following result, which is a direct consequence of Theorem 7. Theorem 8 (Dieudonné–Pick boundary lemma). Let $f\in \mathscr B\{1\}$. Then for each $z\in \mathbb D$ Inequality (2.15) was obtained by Yanagihara [7] and Mercer [8], who did not discuss its sharpness however. We present the most simple proof of (2.15) in our opinion; it is based on Theorem 7. Proof of Theorem 8. Let $f\in \mathscr B\{1\}$. For a fixed point $z_0\in \mathbb D$ consider the composition One can verify that $h$ is a function in $\mathscr B[0,1]$, and its derivatives at the fixed points are
$$
\begin{equation}
h'(0) =\frac{1-z_0}{1-\overline{z}_0}\,\frac{1-\overline{f(z_0)}}{1-f(z_0)}\, \frac{1-|z_0|^2}{1-|f(z_0)|^2}\, f'(z_0)
\end{equation}
\tag{2.17}
$$
and
$$
\begin{equation}
h'(1) =\frac{|1-z_0|^2}{1-|z_0|^2}\,\frac{1-|f(z_0)|^2}{|1-f(z_0)|^2}\,f'(1).
\end{equation}
\tag{2.18}
$$
We substitute (2.17) and (2.18) into inequality (2.12), which holds for $h$ by Theorem 7. Taking the fact that $z_0$ is arbitrary into account we obtain (2.15). The first part of the theorem is proved.
We turn to the proof of the second part. Assume that equality holds in (2.15) for some $ f\in \mathscr B\{1\}$ at some point $z_0\in\mathbb D$. In view of (2.17) and (2.18) this equality can be expressed as follows in terms of the function $h \in \mathscr B[0,1]$ defined by (2.16):
$$
\begin{equation*}
\biggl|h'(0)-\frac{1}{h'(1)}\biggr|= 1-\frac{1}{h'(1)}\,.
\end{equation*}
\notag
$$
However, then by Theorem 7 there exists a point $s\in\mathbb D$ such that either $h(w)=w$ or $h(w)=w L_s (w)$. Hence $f(z)=L^{-1}_{f(z_0)}\circ h\circ L_{z_0}(z)$ is either a linear fractional function or a second-order Blaschke product.
Conversely, let $f(z)=L_q(z)L_p(z)$ for some $q,p\in\mathbb D$. Clearly, $f\in\mathscr B\{1\}$. Substituting the derivatives
$$
\begin{equation*}
\begin{aligned} \, f'(z)&=-\frac{(1-\overline{q})(1-\overline{p})}{(1-q)(1-p)} \\ & \times \frac{q(1-|p|^2)+p(1-|q|^2)-2(1-|q|^2|p|^2)z+ (\overline{q}(1-|p|^2)+\overline{p}(1-|q|^2))z^2} {(1-\overline{q}z)^2(1-\overline{p}z)^2}
\end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
f'(1)=-\frac{(q+\overline{q})(1-|p|^2)+ (p+\overline{p})(1-|q|^2)-2(1-|q|^2|p|^2)}{|1-q|^2|1-p|^2}
\end{equation*}
\notag
$$
into (2.15) we obtain an equality for all $z\in\mathbb D$. In a similar way we obtain equality in (2.15) for $f(z)=L_q(z)$. $\Box$ We can interpret the result of Theorem 8 as a solution of the extremal problem of the exact set of quantities $f'(z_0)$ for the class of functions in $\mathscr B\{1\}$ taking $z_0\in \mathbb D$ to some point $w_0$ and having a fixed angular derivative $f'(1)=\alpha$. By Theorem 8 this set lies in the disc
$$
\begin{equation*}
\biggl|f'(z_0)-\frac{1}{\alpha}\,\frac{(1-w_0)^2}{(1-z_0)^2}\biggr|\leqslant \frac{1-|w_0|^2}{1-|z_0|^2}-\frac{1}{\alpha}\,\frac{|1-w_0|^2}{|1-z_0|^2}\,.
\end{equation*}
\notag
$$
In addition, $f'(z_0)$ lies on the boundary of this disc if and only if either where $q\in\mathbb D$ is such that $f(z_0)=w_0$ and $f'(1)=\alpha$, or where $q,p\in\mathbb D$ are such that $f(z_0)=w_0$ and $f'(1)=\alpha$. Theorem 8 can also be formulated as a refinement of the Julia–Carathéodory theorem in the case when, in addition to the value $f(z)$ of a function $f\in \mathscr B\{1\}$ at some point $z\in\mathbb D$, we also know the value of its derivative $f'(z)$. Theorem 8'. Let $f$ be a function in $\mathscr B\{1\}$ distinct from a linear fractional map of the unit disc $\mathbb D$ to itself. Then for each $z\in \mathbb D$ We can view Theorem 8' as a solution of an extremal problem: on the class of functions in $\mathscr B\{1\}$ that take a point $z_0\in \mathbb D$ to a point $w_0$ and have a fixed derivative $f'(z_0)$, find the greatest lower bound for the angular derivative $f'(1)$. Since $|f'(z_0)|<(1-|w_0|^2)/(1-|z_0|^2)$, given $z_0, w_0\in \mathbb D$ and $c$ such that $|c|<(1-|w_0|^2)/(1-|z_0|^2)$, this problem has the following solution:
$$
\begin{equation*}
\begin{aligned} \, \inf f'(1)&=\frac{|1-w_0|^2}{1-|w_0|^2}\,\frac{1-|z_0|^2}{|1-z_0|^2} \\ &\qquad\times\biggl(1+\biggl|1-c\,\frac{1-|z_0|^2}{1-|w_0|^2}\, \frac{1-z_0}{1-\overline{z}_0}\,\frac{1-\overline{w}_0}{1-w_0}\biggr|^2 \biggl[1-|c|^2\biggl(\frac{1-|z_0|^2}{1-|w_0|^2}\biggr)^2\biggr]^{-1}\biggr), \end{aligned}
\end{equation*}
\notag
$$
where the infimum is taken over all functions $f\in\mathscr B\{1\}$ such that $f(z_0)=w_0$ and $f'(z_0)=c$. The Blaschke product $f(z)=L_q(z)L_p(z)$, where $q,p\in\mathbb D$ are points such that $f(z_0)=w_0$ and $f'(z_0)=c$, is an extremal function here. Proof. By the triangle inequality we have the lower bound
$$
\begin{equation*}
\biggl|f'(z)-\frac{1}{f'(1)}\, \frac{(1-f(z))^2}{(1-z)^2}\biggr|\geqslant \frac{1}{f'(1)}\,\frac{|1-f(z)|^2}{|1-z|^2}-|f'(z)|.
\end{equation*}
\notag
$$
Supplementing it with the upper bound from Theorem 8 we obtain (2.20). $\Box$ Although inequality (2.20) is less sharp than (2.19), it is more compact. The next result is the best possible refinement of Osserman’s inequality (2.11). Corollary 2.2. Let $f\in \mathscr B\{1\}$. Then Moreover, equality holds if and only if where $q/(1-|q|^2)+p/(1-|p|^2)\leqslant 0$. Proof. Inequality (2.21) follows from (2.20) for $z=0$.
We prove the second result. Assume that equality in (2.21) is attained for some $f\in \mathscr B\{1\}$. Then $f(z)=L_q(z)L_p(z)$ by Theorem 8', where $q,p\in\mathbb D$. Now,
$$
\begin{equation}
f(0) =\frac{1-\overline{q}}{1-q}\,\frac{1-\overline{p}}{1-p}\,q p,
\end{equation}
\tag{2.22}
$$
and
$$
\begin{equation}
f'(1) =\frac{1-|q|^2}{|1-q|^2}+\frac{1-|p|^2}{|1-p|^2}\,.
\end{equation}
\tag{2.24}
$$
Thus, we have the equality
$$
\begin{equation*}
\begin{aligned} \, &f'(1)-\frac{2\,|1-f(0)|^2}{1+|f'(0)|-|f(0)|^2} \\ &\qquad=2\biggl(\frac{\operatorname{Re} q}{1-|q|^2}+ \frac{\operatorname{Re} p}{1-|p|^2}+ \biggl|\frac{q}{1-|q|^2}+\frac{p}{1-|p|^2}\biggr|\biggr) |1-q p|^2 \\ &\qquad\qquad\quad\times|1-q|^{-2}|1-p|^{-2} \biggl(\frac{1-|q|^2|p|^2}{(1-|q|^2)(1-|p|^2)}+ \biggl|\frac{q}{1-|q|^2}+\frac{p}{1-|p|^2}\biggr|\biggr)^{-2}, \end{aligned}
\end{equation*}
\notag
$$
whose right-hand side vanishes if and only if $q/(1-|q|^2)+p/(1-|p|^2)\leqslant 0$.
Conversely, let $f(z)=L_q(z)L_p(z)$ for some $q,p\in\mathbb D$ such that $q/(1-|q|^2)+p/(1-|p|^2)\leqslant0$. Then $f\in\mathscr B\{1\}$ and formulae (2.22)–(2.24) hold. Substituting (2.22)–(2.24) into (2.21) and taking the relation between $q$ and $p$ into account we obtain an equality. 3. Conditions for univalence and the geometric properties of holomorphic mapsA holomorphic function $f$ is said to be univalent in a domain $D$ if $f(z_1)\ne f(z_2)$ for all $z_1,z_2\in D$ such that $z_1\ne z_2$. In other words, the equation $f(z)=w$ has at most one root in $D$ for each $w\in \mathbb C$. A univalent function maps $D$ conformally and one-to-one onto a domain in the $w$-plane. Conditions for the univalence of holomorphic maps have both theoretical importance and significance from the point of view of applications. In particular, in the method of complex potential univalence is related to the physical feasibility of a model. In this connection finding sufficient conditions for the univalence of a function in a fixed domain is one of the central problems in geometric function theory. Another classical setting of the problem of univalence is finding domains where the function under consideration is univalent. We focus on this setting. However, for completeness we also present some of the main results on conditions for the univalence in a prescribed domain (for details, see the surveys [9] and [10]). We start with a simple — albeit important for our studies — condition for univalence in an arbitrary convex domain, that is, a domain that contains each line segment connecting two points in the domain. Theorem 9 (Noshiro [11] and Warshawski [12]). A holomorphic function $f$ in a convex domain $D$ is univalent in $D$ if $\operatorname{Re} f'(z)>0$ for all $z\in D$. Proof. Let $z_1,z_2\in D$, $z_1\ne z_2$. As $D$ is convex, $f$ is defined on the line segment $[z_1,z_2]$. Since $\operatorname{Re}f'(z)>0$, it follows that Thus, $f(z_1)\ne f(z_2)$, so that $f$ is univalent in $D$. $\Box$ The question answered most often concerns univalence in a disc or, without loss of generality, in the unit disc $\mathbb D$. A number of sufficient conditions for univalence in $\mathbb D$ are consequences of criteria characterizing maps of $\mathbb{D}$ onto domains with some special geometric properties, which can be expressed as conditions that the real parts of certain functionals are non-negative. We say that a holomorphic function $f$ in $\mathbb D$ is convex in $\mathbb D$ if it is univalent in $\mathbb D$ and maps $\mathbb D$ onto a convex domain. The first mention of convex function was made by Study [13], who gave an analytic description of such functions. Theorem 10 (Study [13]). A holomorphic function $f$ in the disc $\mathbb D$ such that $f(0)=0$ and $f'(0)=1$ is convex in $\mathbb D$ if and only if for each $z\in \mathbb D$. This convexity condition is based on the identity
$$
\begin{equation*}
\frac{d(\pi/2+\theta+\arg f'(re^{i\theta}))}{d\theta}= \operatorname{Re}\biggl(1+z\frac{f''(z)}{f'(z)}\biggr)> 0
\end{equation*}
\notag
$$
for all $z=r e^{i\theta}$, $0<r<1$. Geometrically, it means that the tangent to the curve $L_r=\{w\colon w=f(r e^{i\theta}),\, 0\leqslant \theta\leqslant 2\pi\}$ rotates monotonically anticlockwise as $\theta$ varies from $0$ to $2\pi$. A holomorphic function $f$ in $\mathbb D$ is said to be starlike in $\mathbb D$ if it is univalent in $\mathbb D$ and takes $\mathbb D$ to a domain starlike with respect to the origin, that is, to a domain that contains each line segment connecting a point in it with the origin. An analytic description of starlike functions is due to Alexander [14]. Theorem 11 (Alexander [14]). A holomorphic function $f$ in the disc $\mathbb D$ such that $f(0)=0$ and $f'(0)=1$ is starlike in $\mathbb D$ if and only if for each $z\in \mathbb D$. This starlikeness condition is based on the identity
$$
\begin{equation*}
\frac{d \arg f(re^{i\theta})}{d\theta}= \operatorname{Re}\biggl(z\frac{f'(z)}{f(z)}\biggr)> 0
\end{equation*}
\notag
$$
for all $z=r e^{i\theta}$, $0<r<1$. Geometrically, it means that the line segment $[0,f(re^{i\theta})]$ rotates monotonically anticlockwise as $\theta$ varies from $0$ to $2\pi$. The problem of finding conditions for a function to be starlike in the form of a bound for the modulus of its second derivative $|f''(z)|\leqslant \lambda$, $\lambda>0$, was stated for the first time by Mocanu [15]. He showed that for $\lambda=2/3$ the above inequality implies starlikeness. Subsequently, Ponnusami and Singh [16] improved the constant to $\lambda=2/\sqrt{5}$ . The sharp constant $\lambda=1$ is due to Obradović. Theorem 12 (Obradović [17]). Let $f$ be a holomorphic function in the disc $\mathbb D$ such that $f(0)=0$ and $f'(0)=1$. If $|f''(z)|\leqslant1$ for all $z\in\mathbb D$, then $f$ is starlike in $\mathbb D$. To demonstrate the sharpness of this condition Obradović proposed the example of the function $f(z)=z+(1+\varepsilon)z^2/2$, $\varepsilon>0$, which satisfies $|f''(z)|=1+\varepsilon>1$, but is not even univalent in $\mathbb D$ because its derivative $f'(z)=1+(1+\varepsilon)z$ vanishes at $z=-1/(1+\varepsilon)$. Remark 3.1. The sufficient condition of starlikeness $|f''(z)|\leqslant 1$ is at the same time sufficient for univalence and local univalence. Moreover, the constant $\lambda=1$ is again sharp in this case. Obradović also found the sharp value of the constant ensuring that the function is convex. Theorem 13 (Obradović [17]). Let $f$ be a holomorphic function in $\mathbb D$ such that $f(0)=0$ and $f'(0)=1$. If $|f''(z)|\leqslant1/2$ for all $z\in\mathbb D$, then $f$ is convex in $\mathbb D$. To show that this convexity condition is sharp Obradović considered the function $f(z)=z+(1+\varepsilon)z^2/4$, $\varepsilon\in (0,1)$. It satisfies $|f''(z)|=(1+\varepsilon)/2>1/2$; however, as $1+zf''(z)/f'(z)$ is negative for real $z\to -1$, $f$ is not convex in $\mathbb D$. Špaček [18] extended the concept of a starlike function. A holomorphic fuction $f$ in $\mathbb D$ is said to be spiral-like in $\mathbb D$ if it is univalent in $\mathbb D$ and takes $\mathbb D$ to a spiral-shaped domain of order $\alpha\in(-\pi/2,\pi/2)$, that is, a domain containing the segment of a logarithmic spiral with equation $\operatorname{Im} \{e^{i\alpha}\ln f(z)\}=\operatorname{const}$ that connects any point $f(z)$, $z\in \mathbb D$, with the origin. Theorem 14 (Špaček [18]). A holomorphic function $f$ in $\mathbb D$ such that $f(0)=0$ and $f'(0)=1$ is spiral-like in $\mathbb D$ if and only if for some $\alpha\in(-\pi/2,\pi/2)$, for all $z\in \mathbb D$. This spiral-likeness condition means that $\operatorname{Im}\{e^{i \alpha}\ln f(re^{i\theta})\}$ is an increasing function of $\theta$ for fixed $r$, $0<r<1$, and $\alpha\in(-\pi/2,\pi/2)$. A holomorphic function $f$ in $\mathbb D$ is said to be close to convex in $\mathbb D$ if the image $f(\mathbb D)$ is linearly attainable from outside, that is, the exterior of $f(\mathbb D)$ can be covered by a system of disjoint rays each of which intersects the boundary $\partial f(\mathbb D)$ in a point, a connected line segment, or the whole ray. Theorem 15 (Kaplan [19]). Let $f$ be a holomorphic function in $\mathbb D$ that is locally univalent in $\mathbb D$. Then $f$ is close to convex in $\mathbb D$ if and only if for $z=r e^{i\theta}$, for each $r\in(0,1)$ and all $\theta_1$ and $\theta_2$ such that $\theta_1<\theta_2$. If we assume from the outset that the function is univalent in $\mathbb D$, then it takes some disc with centre zero to a starlike and even convex domain. The precise convexity radius was found by Nevanlinna [20], and the starlikeness radius was found by Grunsky [21]. Theorem 16 (Nevanlinna [20]). If $f$ is a univalent holomorphic function in the disc $\mathbb D$ such that $f(0)=0$ and $f'(0)=1$, then the image of the disc $|z|<2-\sqrt{3}$ is a convex domain. Theorem 17 (Grunsky [21]). If $f$ is a holomorphic univalent function in the disc $\mathbb D$ such that $f(0)=0$ and $f'(0)=1$, then the image of the disc $|z|<\tanh(\pi/4)$ is a starlike domain. An asymptotically sharp estimate for the starlikeness radius under the additional condition that the function is bounded is given by the following theorem (for a sharp estimate, see [22]). Theorem 18 (Goryainov [23]). Let $f$ be a univalent holomorphic function in the disc $\mathbb D$ such that $f(0)=0$, $f'(0)=1$, and $|f(z)|<M$ for all $z\in\mathbb D$, where $M>1$. Then the image of the disc is a starlike domain. Remark 3.2. As $M\to 1$, the starlikeness radius tends to one, and as $M\to \infty$, to $\tanh(\pi/4)$. Conditions for univalence in a domain usually include the requirement that the function be locally univalent, which is a necessary prerequisite for univalence. Finding conditions for the univalence of a locally univalent holomorphic function is closely connected with the concept of Schwarzian derivative,
$$
\begin{equation*}
\{f,z\}=\biggl(\frac{f''(z)}{f'(z)}\biggr)'- \frac{1}{2}\biggl(\frac{f''(z)}{f'(z)}\biggr)^2,
\end{equation*}
\notag
$$
which is known to play an important role in geometric function theory (for instance, see [24] and [25]). We point out the paper [26], containing estimates for the sum of the Schwarzian derivatives at boundary fixed points. The first necessary condition for univalence in terms of the Schwarzian was due to Kraus. However, his result had been ‘forgotten’, and then it was established anew by Nehari. Theorem 19 (Kraus [27] and Nehari [28]). Let $f$ be a holomorphic function in the disc $\mathbb D$ such that $f'(z)\ne 0$ for all $z\in\mathbb D$. If $f$ is univalent in $\mathbb D$, then for $z\in \mathbb D$ We can view (3.1) as a solution to the extremal problem of finding the quantity $\max\bigl(\sup_{z\in \mathbb D}\,|\{f,z\}|(1-|z|^2)^2\bigr)$, where the maximum is taken over all univalent functions in $\mathbb D$. The numerical value of the solution is $6$; it is delivered, for example, by the Koebe function $k(z)={z}(1-z)^{-2}$ with Schwarzian $\{k,z\}=-{6}{(1-z^2)^{-2}}$, $z\in(-1,1)$. Replacing $6$ by $2$ we obtain a sufficient condition for univalence due to Nehari. Theorem 20 (Nehari [28]). Let $f$ be a holomorphic function in the unit disc $\mathbb D$ such that $f'(z)\ne 0$ for all $z\in\mathbb D$. If for all $z\in\mathbb D$, then $f$ is univalent in $\mathbb D$. We can view (3.2) as a solution to the extremal problem of finding the quantity $\inf\bigl(\sup_{z\in \mathbb D}|\{f,z\}|(1-|z|^2)^2\bigr)$, where the infimum is taken over all functions that are locally univalent but not univalent in $\mathbb D$. The numerical value of the solution is $2$, as follows from Hille’s example [29] of the function
$$
\begin{equation*}
f(z)=\biggl(\frac{1-z}{1+z}\biggr)^{i\varepsilon},\qquad \varepsilon>0,
\end{equation*}
\notag
$$
which is not univalent in $\mathbb D$, but has the Schwarzian derivative $\{f,z\}=2(1+\varepsilon^2)(1- z^2)^{-2}$. We also present another sufficient condition for univalence, which has a very simple form. Theorem 21 (Nehari [28]). Let $f$ be a holomorphic function in $\mathbb D$ such that $f'(z)\ne 0$ for all $z\in\mathbb D$. If $|\{f,z\}|\leqslant \pi^2/2$ for $z\in\mathbb D$, then $f$ is univalent in $\mathbb D$. Nehari’s results are based on the connection between a function being univalent and the stability property of solutions of a differential equations of the second order (for more details, see [9]). Based on this connection, Nehari established a general criterion for univalence. Theorem 22 (Nehari [30]). A meromorphic function $f$ in $\mathbb D$ is univalent in $\mathbb D$, if $|\{f,z\}|\leqslant 2p(|z|)$ for all $z\in\mathbb D$, where the majorant $p(x)$ is a continuous non- negative function such that Remark 3.3. Nehari showed that if $p$ is holomorphic in $\mathbb D$ and $|p(z)|\leqslant p(|z|)$, then the condition $|\{f,z\}|\leqslant 2p(|z|)$ is sharp. This general univalence criterion covers, as special cases, Nehari’s results mentioned above, namely Theorems 20 and 21 (which correspond to the majorants $p(x)=(1-x^2)^{-2}$ and $p(x)=\pi^2/4$, respectively), as well as, for example, the following result due to Pokornyi. Theorem 23 (Pokornyi [31]). Let $f$ be a holomorphic function in the unit disc $\mathbb D$ such that $f'(z)\ne 0$ for all $z\in\mathbb D$. If $|\{f,z\}|\leqslant 4(1-|z|^2)^{-1}$ for each $z\in\mathbb D$, then $f$ is univalent in $\mathbb D$. Now we present some univalence tests that have the form of conditions on the pre-Schwarzian $f''/f'$. They can be derived using the theory of Löwner–Kufarev equations (see, for instance, [32] and [33]). Theorem 24 (Duren, Shapiro, and Shields [34]). Let $f$ be a holomorphic function in $\mathbb D$ such that $f'(z)\ne 0$ for all $z\in\mathbb D$. If $f$ is univalent in $\mathbb D$, then for each $z\in\mathbb D$. The Koebe function demonstrates that the constant $6$ is sharp. We note two further results in which conditions on the values of a holomorphic function and its derivatives ensure a homeomorphic extension to $\overline{\mathbb{D}}$, that is, univalence in the closed disc. Some properties of the image of the unit circle are also established on the way. Theorem 25 (Becker [35]). Let $f$ be a holomorphic function in $\mathbb D$ such that $f(0)= 0$, $f'(0)=1$, and for some $\varrho\in(0,1]$ for $z\in\mathbb{D}$. Then $f$ is univalent in $\mathbb{D}$. Moreover, if $\varrho< 1$, then $f(\mathbb{D})$ is a Jordan domain with quasiconformal boundary. Remark 3.4. For $\varrho=1$ Theorem 25 provides a sharp sufficient condition for univalence in $\mathbb D$; this was shown by Becker and Pommerenke [36]. Theorem 26 (Busovskaya and Goryainov [37]). Let $f$ be a holomorphic function in $\mathbb D$ such that $f(0)=0$, $f'(0)=1$, $f(z)/z\ne 0$, and $zf'(z)/f(z)\in G$ for all $z\in\mathbb{D}$, where $G$ is a convex domain satisfying the following conditions: Then $f$ is univalent in $\mathbb{D}$, extends homeomorphically to $\overline{\mathbb{D}}$, and takes $\mathbb{D}$ to a domain bounded by a rectifiable Jordan curve. 4. Fixed points, discs of univalence, and covering discsIf the derivative of $f$ is distinct from zero at an interior point of the domain, then this does not only mean that $f$ is univalent in a neighbourhood of this point. Some neighbourhood of the image of this interior point is fully covered by values of $f$. Moreover, if this neighbourhood is sufficiently small, then this covering has multiplicity one, so that a holomorphic function inverse to $f$ exists in this small neighbourhood. The main question is about how extended neighbourhoods can we take depending on the function in question or the class of functions. Koebe established a well-known theorem on the existence of a disc of absolute radius covered by the values of a univalent function $f$ in the unit disc $\mathbb D$ that is normalized by the conditions $f(0)=0$ and $f'(0)=1$. His conjecture that this absolute radius is $1/4$ was justified by Bieberbach in connection with the coefficient problem for univalent functions. Theorem 27 (Koebe [38] and Bieberbach [39]). If $f$ is a univalent holomorphic function in the disc $\mathbb D$ such that $f(0)=0$ and $f'(0)=1$, then its image $f(\mathbb D)$ contains the disc $|w|<1/4$. Bloch generalized Theorem 27 by showing that if a function is merely holomorphic, then there exists a disc of some absolute radius that is covered univalently by means of this function. Theorem 28 (Bloch [40]). There exists $\beta>0$ with the following property. If $f$ is a holomorphic function in $\mathbb D$ such that $f(0)=0$ and $f'(0)=1$, then there exists a domain $\Delta\subset\mathbb D$ such that $f$ is univalent in it and the image $f(\Delta)$ contains a disc of radius $\beta$. Finding the supremum $B$ of such constants $\beta$, which is called the Bloch constant, is one of the most important and yet unsolved problems in geometric function theory. The lower estimate $B\geqslant \sqrt{3}/4$, which is, in fact, the best known one, is due to Ahlfors [41]. Heins [42] showed that this estimate is not sharp, that is, $B>\sqrt{3}/4$. Bonk [43], who used his distortion theorem for the Bloch class, proved that $B>\sqrt{3}/4+10^{-14}$. Subsequently, Chen and Gauthier [44], who improved slightly some technical details of Bonk’s proof, refined his observation: $B>\sqrt{3}/4+2\cdot 10^{-4}$. An upper estimate for the Bloch constant is due to Ahlfors and Grunsky [45]:
$$
\begin{equation*}
B\leqslant \frac{1}{\sqrt{1+\sqrt{3}}}\, \frac{\Gamma(1/3)\Gamma(11/12)}{\Gamma(1/4)}\,.
\end{equation*}
\notag
$$
There is a conjecture that equality sign holds in the last inequality. On the other hand, in the class of holomorphic functions $f$ in $\mathbb D$ normalized by the conditions $f(0)=0$ and $f'(0)=1$ there exists no universal disc of univalence with centre at zero (this follows from a result of Landau cited below). However, if we restrict this class by adding the condition that functions are bounded by a fixed constant, then the situation changes. Theorem 29 (Landau [46]). Let $f$ be a holomorphic function in the disc $\mathbb D$ such that $f(0)=0$, $f'(0)=1$, and $|f(z)|<M$ for $z\in\mathbb D$, where $M>1$. Then $f$ is univalent in the disc $|z|<M-\sqrt{M^2-1}$ . Moreover, for each $R>M-\sqrt{M^2-1}$ there exists a holomorphic function $f$ in $\mathbb D$ satisfying $f(0)=0$ and $f'(0)=1$ that is bounded by $M$ in $\mathbb D$, but is not univalent in the disc $|z|<R$. Dieudonné and Cacridis-Teodorakopolus discovered remarkable facts: holomorphic functions bounded by a constant $M$ do not only map the disc $|z|<M-\sqrt{M^2-1}$ conformally, but they map it onto starlike (see [1] and [47], Theorem VI.10) and even convex domains [48]. Theorem 30 (Dieudonné [1]). Let $f$ be a holomorphic function in the disc $\mathbb D$ such that $f(0)= 0$ and $f'(0)=1$, and let $|f(z)|<M$ for all $z\in\mathbb D$, where $M>1$. Then $f$ is starlike in the disc $|z|<M-\sqrt{M^2-1}$. Theorem 31 (Cacridis-Teodorakopolus [48]). Let $f$ be a holomorphic function in the disc $\mathbb D$ such that $f(0)= 0$ and $f'(0)=1$, and let $|f(z)|<M$ for all $z\in\mathbb D$, where $M>1$. Then $f$ is a convex function in the disc $|z|<M-\sqrt{M^2-1}$. On the basis of Theorem 29 Landau found the best possible disc covered univalently by all holomorphic functions $f$ in $\mathbb D$ normalized by the conditions $f(0)=0$ and $f'(0)=1$. Theorem 32 (Landau [46]). Let $f$ be a holomorphic function in the disc $\mathbb D$ such that $f(0)= 0$ and $f'(0)=1$, and let $|f(z)|<M$ for some $M>1$ for all $z\in\mathbb D$. Then there exists a function inverse to $f$ that maps the disc $|w|<M(M-\sqrt{M^2-1}\,)^2$ conformally onto a domain $\mathscr X\subset \mathbb D$. Moreover, for each $R>M(M-\sqrt{M^2-1}\,)^2$ there exists a holomorphic function $f$ in $\mathbb D$ satisfying $f(0)=0$ and $f'(0)=1$ and bounded by $M$ in $\mathbb D$ that has no inverse function in the disc $|w|<R$. Theorem 32 underlies one of the first lower estimates for the Bloch constant. In what follows we present the slightly modified proofs of both theorems of Landau’s. We also use the ideas of these proofs below, to find sharp domains of univalence and univalent covering domains for the class of functions with an interior and a boundary fixed point. If a bounded function is also univalent, then Koebe’s covering theorem (Theorem 27) can be improved as follows. Theorem 33 (Pick [49]). Let $f$ be a univalent holomorphic function in the disc $\mathbb D$ such that $f(0)= 0$ and $f'(0)=1$, and let $|f(z)|<M$ for all $z\in\mathbb D$, where $M>1$. Then its image $f(\mathbb D)$ contains the disc $|w|<M(2M-1-2\sqrt{M(M-1)}\,)$. Remark 4.1. As $M\to 1$, the radius of the covered disc tends to $1$, while as $M\to \infty$, it tends to $1/4$. We turn to the proofs of Landau’s theorems. First we formulate his result on a domain of univalence (Theorem 29) in terms of self-maps of the unit disc. For any $M>1$ we distinguish the subclass $\mathscr B_M[0]$ of $\mathscr B[0] $ consisting of functions such that the moduli of their derivatives at $z=0$ are bounded away from zero by $1/M$:
$$
\begin{equation*}
\mathscr B_{M}[0]=\biggl\{f\in \mathscr B[0]\colon |f'(0)|\geqslant \frac{1}{M}\biggr\}.
\end{equation*}
\notag
$$
Theorem 29'. Let $f\in \mathscr B_M[0]$, $M>1$. Then $f$ is univalent in the disc For any domain $\mathscr{V}$ such that $\mathscr L\subset\mathscr{V}\subset\mathbb D$ and $\mathscr{V}\ne \mathscr L$ there exists $f\in \mathscr B_M[0]$ that is not univalent in $\mathscr{V}$. The proof of Theorem 29' is based on Landau’s inequality which estimates the common value that a function can take at two distinct points. Lemma 4.1 (Landau [46]). Let $f\in \mathscr B[0]$, and let $a, b\in \mathbb D$, $a\ne b$, be points such that $f(a)=f(b)=c$. Then Proof. Given $f\in \mathscr B[0]$, consider the linear fractional transformation It is obvious that $g\in \mathscr B$; furthermore, $g(a)=g(b)=0$. Then by the Schwarz–Pick lemma $g$ can be represented in the form
$$
\begin{equation*}
g(z)=\frac{z-a}{1-\overline{a}z}\,\frac{z-b}{1-\overline{b}z}\,h(z),
\end{equation*}
\notag
$$
where either $h\in \mathscr B$, or $h$ is an identical constant of modulus at most one. Setting $z=0$ in this representation we obtain $c=a b \,h(0)$. This yields (4.1). $\Box$ Inequality (4.1) is sharp, and equality is attained at Blaschke products. Note that a generalization of (4.1) for $n$ distinct points was obtained in [50]. Lemma 4.2 (Löwner [51]). Let $f\in \mathscr B[0]$ and $f(z)\not\equiv \varkappa z$, $|\varkappa|=1$. Then for each $r$, $0<r\leqslant 1$, the image of the disc $|z|\leqslant r$ under the map $w(z)=f(z)/z$ lies in the non-Euclidean disc Proof. Since $f\in \mathscr B[0]$, by Schwarz’s lemma we have $w(z)=f(z)/z\in \mathscr B$, and $w(0)=f'(0)$. Then by the Schwarz–Pick lemma
$$
\begin{equation*}
\biggl|\frac{w(z)-f'(0)}{1-\overline{f'(0)}w(z)}\biggr|\leqslant |z|
\end{equation*}
\notag
$$
for each $z\in \mathbb D$. Hence, if $|z|\leqslant r$, $0<r\leqslant 1$, then $w(z)$ lies in the non-Euclidean disc (4.2). $\Box$ Proof of Theorem 29'. We leave out the case $f(z)\equiv \varkappa z$, $|\varkappa|=1$, which is obvious. Let $f\in \mathscr B_M[0]$, $M>1$, be not univalent in $\mathscr L$, that is, assume that there exist points $z_1, z_2\in \mathscr L$, $z_1\ne z_2$, such that $f(z_1)=f(z_2)$. Then $|f(z_1)|\leqslant |z_1|\, |z_2|<(M-\sqrt{M^2-1})\,|z_1|$ by Lemma 4.1, which is equivalent to On the other hand, by Lemma 4.2 the function $w(z)=f(z)/z$ maps $\mathscr L$ to the non-Euclidean disc
$$
\begin{equation*}
\biggl|\frac{w-f'(0)}{1-\overline{f'(0)}w}\biggr|\leqslant M-\sqrt{M^2-1}\,,
\end{equation*}
\notag
$$
that is, to the Euclidean disc $|w-c|\leqslant R$, where
$$
\begin{equation*}
c=\frac{1-(M-\sqrt{M^2-1}\,)^2}{1-(M-\sqrt{M^2-1}\,)^2\,|f'(0)|^2}\,f'(0) \end{equation*}
\notag
$$
and
$$
\begin{equation*}
R=(M-\sqrt{M^2-1}\,)\,\frac{1-|f'(0)|^2}{1-(M-\sqrt{M^2-1}\,)^2\,|f'(0)|^2}\,.
\end{equation*}
\notag
$$
For each $w$ in this disc we have
$$
\begin{equation}
|w|\geqslant |c|-R= \frac{|f'(0)|-(M-\sqrt{M^2-1}\,)}{1-(M-\sqrt{M^2-1}\,)|f'(0)|}\,.
\end{equation}
\tag{4.4}
$$
From (4.3) and (4.4) we obtain which is equivalent to the inequality $|f'(0)|<1/M$, which contradicts the assumption that $f$ belongs to $\mathscr B_M[0]$.
To prove the second part of the theorem, for each point on the boundary of $\mathscr L$ it is sufficient to produce a function in $\mathscr B_M[0]$ whose derivative vanishes at this point. For any $\varkappa$, $|\varkappa|=1$, consider the function It can readily be verified that for $\varkappa$ such that $|\varkappa|=1$ the function $f_\varkappa$ belongs to $\mathscr B_M[0]$ and its derivative has a zero at $z_{\varkappa}=\varkappa(M-\sqrt{M^2-1}\,)$. $\Box$Now we state and prove Landau’s theorem on a univalent covering domain (Theorem 32) for the class $\mathscr B_M[0]$. Theorem 32'. Let $f\in \mathscr B_M[0]$, where $M>1$. Then there exists an inverse function of $f$ that maps the disc conformally onto a domain $\mathscr X\subset \mathbb D$. For any domain $\mathscr V$ such that $\mathscr W\subset \mathscr V \subset \mathbb D$ and $\mathscr V\ne \mathscr W$ there exists $f\in \mathscr B_M[0]$ that has no inverse function in $\mathscr V$. Proof. Let $f\in \mathscr B_M[0]$, where $M>1$. To prove the theorem it is sufficient to verify that for each $w\in \mathscr W$ the equation $ f(z)=w$ has a unique solution in some subdomain of $\mathbb D$. We show that we can take the disc $\mathscr L=\{z\in \mathbb D\colon |z|<M-\sqrt{M^2-1}\,\}$ as such a domain. For $z$ on the circle $|z|=M-\sqrt{M^2-1}$ the values of $g(z)=f(z)/z$ lie in the non-Euclidean disc
$$
\begin{equation*}
\biggl|\frac{g(z)-f'(0)}{1-\overline{f'(0)}g(z)}\biggr|\leqslant M-\sqrt{M^2-1}
\end{equation*}
\notag
$$
by Lemma 4.2. Just as in the proof of Theorem 29', we deduce the lower estimate Taking the inequality $|f'(0)|\geqslant 1/M$ into account we obtain $|g(z)|\geqslant M-\sqrt{M^2-1}$ . Finally, recalling that $g(z)=f(z)/z$ and $|z|=M-\sqrt{M^2-1}$ we conclude that $|f(z)|\geqslant (M-\sqrt{M^2-1}\,)^2$ for all $z$ on the circle $|z|=M-\sqrt{M^2-1}$ .
Thus, as $z$ makes one turn along the circle $|z|=M-\sqrt{M^2-1}$ , the argument of $f(z)-w$ increases by $2\pi$ for any point $w$ in $\mathscr W$. Hence the equation $f(z)=w$ has a unique solution in $\mathscr L$. We turn to the proof of the second part of the theorem. For each $\varkappa$ such that $|\varkappa|=1$ we look at the function It belongs to the class $\mathscr B_M[0]$, and its derivative has a zero at $z_{\varkappa}=\varkappa\bigl(M-\sqrt{M^2-1}\,\bigr)$. Hence the inverse function of $f_\varkappa$ has a branch point at Thus, each point on the boundary of the disc $\mathscr W$ is an obstacle to extending the domain of invertibility. $\Box$5. Fixed points and integral representationsIn this section we obtain integral representations for classes of functions with two fixed points. These will be our tools for finding estimates for domains of univalence and Taylor coefficients. Let $\mathscr C$ denote the class of holomorphic functions $h$ in $\mathbb D$ such that $\operatorname{Re}h(z)>0$ for $z\in \mathbb D$. By the Riesz–Herglotz theorem (for instance, see [32]) a function $h$ belongs to $\mathscr C$ if and only if
$$
\begin{equation}
h(z)=\int_{\mathbb T}\frac{1+\varkappa z}{1-\varkappa z}\,d\nu(\varkappa)+iv,
\end{equation}
\tag{5.1}
$$
where $v=\operatorname{Im}h(0)$, $\nu$ is a positive Borel measure on $\mathbb T$ with total mass $\nu(\mathbb T)=\operatorname{Re}h(0)$. Bearing in mind the bijective correspondence $f\leftrightarrow h$ between the classes $\mathscr B$ and $\mathscr C$ which is established by the equalities
$$
\begin{equation*}
h(z)=\frac{1+f(z)}{1-f(z)}\quad\text{and}\quad f(z)=\frac{h(z)-1}{h(z)+1}\,,
\end{equation*}
\notag
$$
in [6] the representation (5.1) was specialized for subclasses of $\mathscr B$ containing functions with two fixed points. As before, let $\mathscr B[0,1]$ denote the class of functions $f$ in $\mathscr B$ that fix the origin and the boundary point $z=1$. Lemma 5.1 (Goryainov [6]). A function $f$ belongs to the class $\mathscr B[0,1]$ and has an angular derivative $f'(1)=\alpha$, $\alpha>1$, if and only if the function $h(z)=(1+ f(z))/(1-f(z))$ has a representation where $\mu$ is a probability measure on the unit circle $\mathbb T$ such that $\mu(\{1\})=0$. Proof. Let $f\in\mathscr B[0,1]$, and let $f'(1)=\alpha>1$. Then
$$
\begin{equation*}
\frac{1+f(z)}{1-f(z)}= \int_{\mathbb T}\frac{1+\varkappa z}{1-\varkappa z}\,d\nu(\varkappa),
\end{equation*}
\notag
$$
where $\nu$ is a probability measure on $\mathbb T$. In addition,
$$
\begin{equation*}
\frac{1}{\alpha}=\lim_{ x\to 1}\frac{(1-x)(1+f(x))}{(1+x)(1-f(x))}= \lim_{x\to 1}\int_{\mathbb T} \frac{(1-x)(1+\varkappa x)}{(1+x)(1-\varkappa x)}\,d\nu(\varkappa).
\end{equation*}
\notag
$$
Since for all $x\in(0,1)$ and $\varkappa \in \mathbb T$ we have
$$
\begin{equation*}
\biggl|\frac{(1-x)(1+\varkappa x)}{(1+x)(1-\varkappa x)}\biggr|\leqslant 1,
\end{equation*}
\notag
$$
by Lebesgue’s dominated convergence theorem we can take the limit under the integral sign. Note that
$$
\begin{equation*}
\lim_{x\to 1}\frac{(1-x)(1+\varkappa x)}{(1+x)(1-\varkappa x)}=\begin{cases} 0 & \text{for}\ \varkappa\ne 1, \\ 1 & \text{for}\ \varkappa= 1, \end{cases}
\end{equation*}
\notag
$$
so we arrive at the equality
$$
\begin{equation*}
\frac{1}{\alpha}=\lim_{x\to 1}\frac{(1-x)(1+f(x))}{(1+x)(1-f(x))}=\nu(\{1\}).
\end{equation*}
\notag
$$
Hence the function $h(z)=(1+f(z))/(1-f(z))$ has the representation
$$
\begin{equation*}
h(z)=\frac{1}{\alpha}\,\frac{1+z}{1-z}+\int_{\mathbb T \setminus\{1\}} \frac{1+\varkappa z}{1-\varkappa z}\,d\nu(\varkappa).
\end{equation*}
\notag
$$
Replacing $\nu$ by the measure $\mu$ using the equalities
$$
\begin{equation*}
d\mu(\varkappa)=\frac{\alpha}{\alpha-1}\,d\nu(\varkappa) \quad\text{for}\, \varkappa\in\mathbb T \setminus\{1\}\quad\text{and}\quad \mu(\{1\})=0,
\end{equation*}
\notag
$$
we arrive at (5.2).
Conversely, if $\mu$ is a probability measure on $\mathbb T$ such that $\mu(\{1\})=0$ and $\alpha>1$, then (5.2) defines a holomorphic function with positive real part in $\mathbb D$ such that $h(0)=1$. Moreover, $\lim_{x\to 1}h(x)=\infty$. This means that the function $f(z)=(h(z)-1)/(h(z)+1)$ belongs to the class $\mathscr B[0,1]$. In addition,
$$
\begin{equation*}
\begin{aligned} \, \frac{1}{f'(1)}&=\lim_{x\to 1}\frac{(1-x)(1+f(x))}{(1+x)(1-f(x))}= \lim_{x\to 1}\frac{1-x}{1+x}\,h(x) \\ &=\frac{1}{\alpha}+\frac{\alpha-1}{\alpha}\lim_{x\to 1}\int_{\mathbb T} \frac{(1-x)(1+\varkappa x)}{(1+x)(1-\varkappa x)}\,d\mu(\varkappa)= \frac{1}{\alpha}+\frac{\alpha-1}{\alpha}\mu(\{1\})=\frac{1}{\alpha}\,. \end{aligned}
\end{equation*}
\notag
$$
Thus, $f(1)=1$ and $f'(1)=\alpha$. $\Box$ Let $\mathscr B\{-1,1\}$ denote the set of functions $f$ in $\mathscr B$ that fix the boundary points $z=\pm 1$ and have finite angular derivatives $f'(1)$ and $f'(-1)$. Lemma 5.2 (Goryainov [6]). A function $f$ belongs to the class $\mathscr B\{-1,1\}$ if and only if the function $h(z)=(1+f(z))/(1-f(z))$ has a representation of the form where $\rho>0$, $\lambda\in(0,1)$, and $\mu$ is a probability measure on the unit circle $\mathbb T$ such that $\mu(\{-1,1\})=0$. In addition, $f'(-1)=\rho$ and $f'(1)f'(-1)=1/\lambda$. Proof. Let $f\in\mathscr B\{-1,1\}$ and assume that the conditions $f'(-1)=\rho$ and $f'(1)f'(-1)=1/\lambda$ are satisfied. Then $h(z)=(1+f(z))/(1-f(z))$ is a holomorphic function with positive real part in $\mathbb D$. Hence there exists a positive Borel measure $\nu$ on $\mathbb T$ such that
$$
\begin{equation*}
h(z)=\int_{\mathbb T}\frac{1+\varkappa z}{1-\varkappa z}\,d\nu(\varkappa)+iv,
\end{equation*}
\notag
$$
where $v=\operatorname{Im} h(0)$ and $\nu(\mathbb T)=\operatorname{Re} h(0)$. As in the proof of Lemma 5.1,
$$
\begin{equation*}
\lambda \rho=\frac{1}{f'(1)}=\lim_{x\to 1}\frac{(1-x)(1+f(x))}{(1+x)(1-f(x))}= \lim_{x\to 1}\frac{1-x}{1+x}h(x)=\nu(\{1\}).
\end{equation*}
\notag
$$
Now,
$$
\begin{equation*}
\begin{aligned} \, \rho= f'(-1)&=\lim_{x\to -1}\frac{(1-x)(1+f(x))}{(1+x)(1-f(x))}= \lim_{x\to -1}\frac{1-x}{1+x}h(x)=\lim_{x\to -1}\operatorname{Re} \biggl(\frac{1-x}{1+x}h(x)\biggr) \\ &=\lim_{x\to -1}\int_{\mathbb T}\operatorname{Re} \biggl(\frac{(1-x)(1+\varkappa x)}{(1+x)(1-\varkappa x)}\biggr)\, d\nu(\varkappa)=\lim_{x\to -1}\int_{\mathbb T} \frac{(1-x)^2}{|1-\varkappa x|^2}\,d\nu(\varkappa). \end{aligned}
\end{equation*}
\notag
$$
We note that
$$
\begin{equation*}
\frac{d}{dx}\,\frac{(1-x)^2}{|1-\varkappa x|^2}= \frac{2(1-x^2)}{|1-\varkappa x|^4}(\operatorname{Re}\varkappa-1)\leqslant 0
\end{equation*}
\notag
$$
for all $\varkappa\in\mathbb T$ and $x\in(-1,1)$. Then by the monotone convergence theorem we have
$$
\begin{equation}
\lim_{x\to -1}\int_{\mathbb T} \frac{(1-x)^2}{|1-\varkappa x|^2}\,d\nu(\varkappa)= \int_{\mathbb T}\frac{4\,d\nu(\varkappa)}{|1+\varkappa|^2}=\rho.
\end{equation}
\tag{5.4}
$$
Hence it follows, in particular, that $\nu(\{-1\})=0$. However, then $h$ can be represented in the form
$$
\begin{equation*}
h(z)=\lambda\rho\,\frac{1+z}{1-z}+\int_{\mathbb T\setminus \{-1,1\}} \frac{1+\varkappa z}{1-\varkappa z}\,d\nu(\varkappa)+iv.
\end{equation*}
\notag
$$
Since the angular derivative $f'(-1)$ is real, it follows that
$$
\begin{equation*}
\begin{aligned} \, 0&=\operatorname{Im}f'(-1)=\lim_{x\to -1}\operatorname{Im} \biggl(\frac{1-x}{1+x}h(x)\biggr) \\ &=\lim_{x\to -1}\frac{1-x}{1+x} \biggl(\int_{\mathbb T\setminus \{-1,1\}}\operatorname{Im} \frac{1+\varkappa x}{1-\varkappa x}\,d\nu(\varkappa)+v\biggr). \end{aligned}
\end{equation*}
\notag
$$
Therefore,
$$
\begin{equation*}
v=-\lim_{x\to -1}\int_{\mathbb T\setminus \{-1,1\}}\operatorname{Im} \frac{1+\varkappa x}{1-\varkappa x}\,d\nu(\varkappa)= i \lim_{x\to -1}\int_{\mathbb T\setminus \{-1,1\}} \frac{x(\varkappa-\overline{\varkappa})}{|1-\varkappa x|^2}\,d\nu(\varkappa).
\end{equation*}
\notag
$$
If $x\in(-1,-1/2)$ and $\varkappa \in \mathbb T\setminus\{-1,1\}$, then
$$
\begin{equation*}
|1-\varkappa x|^2=(1+x)^2-2x(1+\operatorname{Re} \varkappa)> 1+\operatorname{Re}\varkappa=\frac{|1+\varkappa|^2}{2}\,,
\end{equation*}
\notag
$$
which takes us to the inequality
$$
\begin{equation*}
\frac{|x(\varkappa -\overline{\varkappa})|}{|1-\varkappa x|^2}\leqslant \frac{4}{|1+\varkappa|^2}\,.
\end{equation*}
\notag
$$
Hence we can use Lebesgue’s dominated convergence theorem, so that
$$
\begin{equation*}
iv=\int_{\mathbb T\setminus \{-1,1\}} \frac{\varkappa-\overline{\varkappa}}{|1+\varkappa|^2}\,d\nu(\varkappa).
\end{equation*}
\notag
$$
Thus, the representation for $h$ becomes
$$
\begin{equation}
\begin{aligned} \, h(z)&=\lambda\rho\,\frac{1+z}{1-z}+\int_{\mathbb T\setminus \{-1,1\}} \biggl(\frac{1+\varkappa z}{1-\varkappa z}+ \frac{\varkappa-\overline{\varkappa}}{|1+\varkappa|^2}\biggr)\,d\nu(\varkappa) \notag \\ &=\lambda\rho\,\frac{1+z}{1-z}+2(1+z)\int_{\mathbb T\setminus \{-1,1\}} \frac{1+\varkappa}{(1-\varkappa z)|1+\varkappa|^2}\,d\nu(\varkappa). \end{aligned}
\end{equation}
\tag{5.5}
$$
Recalling that $\nu(\{-1\})=0$, $\nu(\{1\})=\rho\lambda$, from (5.4) we obtain
$$
\begin{equation*}
\rho=\int_{\mathbb T}\frac{4\,d\nu(\varkappa)}{|1+\varkappa|^2}= \rho\lambda+\int_{\mathbb T\setminus\{-1,1\}} \frac{4\,d\nu(\varkappa)}{|1+\varkappa|^2}\,,
\end{equation*}
\notag
$$
and therefore
$$
\begin{equation*}
\int_{\mathbb T\setminus\{-1,1\}} \frac{4\,d\nu(\varkappa)}{|1+\varkappa|^2}=\rho(1-\lambda).
\end{equation*}
\notag
$$
Hence the measure $\mu$ on $\mathbb T\setminus \{-1,1\}$ defined by
$$
\begin{equation*}
d\mu(\varkappa)=\frac{4\,d\nu(\varkappa)}{\rho(1-\lambda)|1+\varkappa|^2}\,,
\end{equation*}
\notag
$$
is a probability measure. Setting $\mu(\{-1,1\})=0$, we obtain a probability measure on $\mathbb T$. In terms of it, the representation (5.5) has the form (5.3).
Conversely, let $h$ be a function with representation (5.3). Since
$$
\begin{equation*}
\operatorname{Re}\frac{(1+z)(1+\varkappa)}{1-\varkappa z}= \frac{(1-|z|^2)(1+\operatorname{Re}\varkappa)}{|1-\varkappa z|^2}\geqslant 0
\end{equation*}
\notag
$$
for all $z\in\mathbb D$ and $\varkappa \in \mathbb T$, it follows that $\operatorname{Re} h(z)>0$ for $z\in\mathbb D$. This means that $f(z)=(h(z)-1)/(h(z)+1)$ is a function in $\mathscr B$. We observe that $\lim_{x\to 1}h(x)=\infty$ and $\lim_{x\to -1}h(x)=0$, and conclude that $f$ fixes the points $z=\pm 1$. From (5.3) we obtain
$$
\begin{equation*}
\frac{(1-x)(1+f(x))}{(1+x)(1-f(x))}=\lambda\rho+ \frac{(1-\lambda)\rho}{2}\int_{\mathbb T} \frac{(1-x)(1+\varkappa)}{1-\varkappa x}\,d\mu(\varkappa),
\end{equation*}
\notag
$$
so that
and Thus, $f\in \mathscr B\{-1,1\}$, $f'(-1)=\rho$, and $f'(1)=1/(\lambda\rho)$. $\Box$ From Lemma 5.2 we deduce an integral representation for the subclass $\mathscr D\{-1,1\}$ of functions preserving the real diameter:
$$
\begin{equation*}
\mathscr D\{-1,1\}=\bigl\{f\in \mathscr B\{-1,1\}\colon f\bigl((-1,1)\bigr)=(-1,1)\bigr\}.
\end{equation*}
\notag
$$
Lemma 5.3 (Kudryavtseva [52]). A function $f$ belongs to the class $\mathscr D\{-1,1\}$ if and only if $h(z)=(1+f(z))/(1-f(z))$ has a representation of the form where $\rho>0$, $\lambda\in(0,1)$, and $\mu$ is a probability measure on the unit circle $\mathbb T$ such that $\mu(\{-1,1\})=0$. In this case $f'(-1)=\rho$ and $f'(1)f'(-1)=1/\lambda$. 6. Boundary fixed points and related domains of univalenceThe fact that the derivative at an interior fixed point is distinct from zero is necessary and sufficient for local univalence. In the case of a boundary fixed point we see from the preceding sections that the angular derivative is distinct from zero. Thus it is natural to expect that we have univalence in a neighbourhood of this point, although the concept of a neighbourhood of a boundary point must itself be clarified. Valiron [53] noted that this occurs indeed, provided that the angular derivative at the boundary fixed point is finite. Theorem 34 (Valiron [53]). Let $f\in\mathscr B\{1\}$ and $f'(1)<\infty$. Then for each $\varphi\in(0,\pi/2)$ there exists a sufficiently small $r>0$ such that $f$ is univalent in the sector Thus he showed that, as concerns local univalence, angular derivatives are not much less potent than ordinary ones. While a function with non-zero derivative is univalent in a disc, a function with finite angular derivative is univalent in a sector with opening arbitrarily close to $\pi$. Of course, the radius of this sector depends not only on the opening, but also on the function itself. Becker and Pommerenke [54], [55] examined thoroughly the shape and sizes of domains of univalence of a function $f\in\mathscr B\{1\}$ in their dependence on the behaviour of the iterates of $f$. It turns out, in particular, that certain conditions ensure that univalence does not only take place in a sector, but even in a domain tangent to the unit circle at $z=1$. There is also some analogy with interior points in problems of the description of domains of univalence of particular functions. Given a function $f\in \mathscr B\{1\}$ with finite angular derivative at a boundary fixed point, Becker and Pommerenke [56] indicated a domain where $f$ is univalent. Theorem 35 (Becker and Pommerenke [56]). Let $f\in \mathscr B\{1\}$ and let $f'(1)<\infty$. Then $f$ is univalent in the domain The proof of Theorem 35 is based on the Becker–Pommerenke inequality (6.3), which is analogous to Landau’s inequality (4.1) in a certain sense. We prove this inequality using ideas due to Landau. Lemma 6.1 (Becker and Pommerenke [56]). Let $f\in \mathscr B\{1\}$, and let $a, b\in \mathbb D$, $a\ne b$, be points such that $f(a)=f(b)=c$. Then Proof. Let $f\in \mathscr B\{1\}$. Then the function
$$
\begin{equation*}
g(z)=\frac{1-\overline{c}}{1-c}\,\frac{f(z)-c}{1-\overline{c}f(z)}
\end{equation*}
\notag
$$
is in the class $\mathscr B\{1\}$ and $g(a)=g(b)=0$. By the Schwarz–Pick lemma $g$ has the following representation:
$$
\begin{equation*}
g(z)=\frac{1-\overline{a}}{1-a}\,\frac{z-a}{1-\overline{a}z}\, \frac{1-\overline{b}}{1-b}\,\frac{z-b}{1-\overline{b}z}\,h(z),
\end{equation*}
\notag
$$
where either $h\in \mathscr B\{1\}$ or $h(z)\equiv 1$. If $h(z)\equiv 1$, then equality holds in (6.3). If $h\in \mathscr B\{1\}$, then the function has a positive angular derivative at $z=1$. On the other hand A generalization of (6.3) to $n$ distinct points was obtained in [50]. Proof of Theorem 35. Note that the domain $\mathscr P$ (see (6.2)) is not empty because by the Julia–Carathéodory theorem the left-hand side of (6.2) is always greater than 1 and certainly less than 2 in a Stolz angle of sufficiently small radius, with opening arbitrarily close to $\pi$. If $f$ is not univalent in $\mathscr P$, then there exist points $a,b\in \mathscr P$, $a\ne b$, such that $f(a)=f(b)=c$. Since the inequality in (6.2) holds at both $a$ and $b$, namely,
$$
\begin{equation*}
\begin{gathered} \, f'(1)\,\frac{1-|c|^2}{|1-c|^2}<2\,\frac{1-|a|^2}{|1-a|^2}\quad\text{and}\quad f'(1)\,\frac{1-|c|^2}{|1-c|^2}<2\,\frac{1-|b|^2}{|1-b|^2}\,, \end{gathered}
\end{equation*}
\notag
$$
we see that
$$
\begin{equation*}
f'(1)\,\frac{1-|c|^2}{|1-c|^2}<\frac{1-|a|^2}{|1-a|^2}+ \frac{1-|b|^2}{|1-b|^2}\,.
\end{equation*}
\notag
$$
This contradicts (6.3). $\Box$ Remark 6.1. By the Julia–Carathéodory theorem the left-hand side of the inequality in (6.2) is always greater than 1. Moreover, it tends to zero in any angle, so that it is less than 2 in the part of a sufficiently small neighbourhood that lies inside such an angle. Thus, Theorem 35 provides a quantitative specification of Valiron’s result on local univalence in a neighbourhood of a boundary fixed point (see Theorem 34). For the class of functions satisfying certain conditions on the derivative at a boundary fixed point the situation with distinguishing domains of univalence is drastically different from the case of an interior point. As shown in [57], even when the sector (6.1) has a small opening, in general we cannot find a universal value of the radius of a sector of univalence if we know only the value of the angular derivative at the boundary point. For any $\alpha>1$ in $\mathscr B\{1\}$ we distinguish the subclass $\mathscr B_{\alpha}\{1\}$ of functions satisfying a constraint on the value of the angular derivative at the fixed point:
$$
\begin{equation*}
\mathscr B_{\alpha}\{1\}= \{f\in \mathscr B\{1\}\colon f'(1)\leqslant \alpha\}.
\end{equation*}
\notag
$$
Theorem 36 (Kudryavtseva and Solodov [57]). For no $\alpha>1$ do there exist non- empty domains of univalence for the class $\mathscr B_{\alpha}\{1\}$. In other words, the class $\mathscr B_{\alpha}\{1\}$ is too wide, and we must consider narrower classes in order that a non-trivial domain of univalence might exist. The most natural reduction of the class $\mathscr B_\alpha\{1\}$, $\alpha>1$, is its subclass of functions that, apart from the repelling fixed point $z=1$, also have a prescribed attracting fixed point (whose existence was established by Denjoy and Wolf; see § 7 for details). On the one hand such a reduction is fairly natural and, moreover, $\mathscr B_\alpha\{1\}$ falls into classes of functions with two fixed points. On the other hand classes of functions with several (particularly, two) fixed points are prolific for various extremal problems and important for applications. First we look at the case when the additional fixed point lies in the interior. Without loss of generality we assume that this is the origin. Set
$$
\begin{equation*}
\mathscr B_\alpha[0,1]=\mathscr B_\alpha\{1\}\cap \mathscr B[0],\qquad \alpha>1.
\end{equation*}
\notag
$$
For $\alpha\in(1,2)$ Theorem 7 implies that $\mathscr B_{\alpha}[0,1]$ is embedded in $\mathscr B_{\alpha/(2-\alpha)}[0]$. From this embedding and Theorem 29' we deduce the following result. Theorem 37 (Goryainov [6]). Let $f\in \mathscr B_{\alpha}[0,1]$, where $\alpha\in (1,2)$. Then $f$ is univalent in the disc Thus, for each $\alpha\in(1,2)$ there is a non-empty domain of univalence for the class $\mathscr B_{\alpha}[0,1]$. Moreover, among all discs with centre at the origin in which all functions in $\mathscr B_{\alpha}[0,1]$ are univalent, $\mathscr{O}(\alpha)$ has the largest radius. On the other hand $\mathscr B_{\alpha}[0,1]$ is a much narrower class than $\mathscr B_{\alpha/(2-\alpha)}[0]$, so there is a prospect of increasing the domain of univalence $\mathscr{O}(\alpha)$. The first step in this direction was made in [6], where it was discovered that the domain of univalence can be extended in the case of the class $\mathscr B_{\alpha}[0,1]$. Theorem 38 (Goryainov [6]). Let $f\in \mathscr B_{\alpha}[0,1]$, where $\alpha\in(1,2)$. Then $f$ is univalent in the domain Proof. Let $f\in \mathscr B_{\alpha}[0,1]$, $\alpha\in(1,2)$. By Lemma 5.1 the function $h(z)=(1+f(z))/(1-f(z))$ has a representation (5.2). We set $\zeta=L(z)=(1+z)/(1-z)$ and consider the function $g(\zeta)=h\circ L^{-1}(\zeta)$. It is holomorphic in the right-hand half-plane $\mathbb H=\{\zeta\in \mathbb C\colon\operatorname{Re}\zeta>0\}$ and has the form
The real part of the derivative of $g$ has a lower bound:
$$
\begin{equation*}
\operatorname{Re} g'(\zeta)\geqslant\frac{1}{\alpha}+ \biggl(1-\frac{1}{\alpha}\biggr)\min_{\varkappa\in\mathbb T} \operatorname{Re}\frac{4\varkappa}{((1-\varkappa)\zeta+1+\varkappa)^2}\,,
\end{equation*}
\notag
$$
which shows that the condition $\operatorname{Re}g'(\zeta)>0$ holds for $\zeta\in L(\mathscr A(\alpha))$. Since $L(\mathscr A(\alpha))$ is a convex domain, $g$ is univalent in $L(\mathscr A(\alpha))$ by the Noshiro–Warshawski theorem (see Theorem 9). But this means that $f=L^{-1}\circ g\circ L$ is univalent in $\mathscr A(\alpha)$. $\Box$ For all $\alpha\in(1,2)$ the domains $\mathscr A(\alpha)$ contain an interior fixed point and adjoin the boundary fixed point, so that these fixed points are connected by a ‘passage of univalence’. From this point of view $\alpha=2$ is a critical value: the function $f(z)=z^2$ belongs to $\mathscr B_{2}[0,1]$, but it is not univalent in any neighbourhood of zero. Theorem 38 was refined in [57]: for $\alpha\in(1,2)$ the domains of univalence were extended, and for $\alpha\in[2,4)$, that is, for $\alpha$ greater than the critical value, domains of univalence were discovered. Theorem 39 (Kudryavtseva and Solodov [57]). Let $f\in \mathscr B_{\alpha}[0,1]$, where $\alpha \in (1,4)$. Then $f$ is univalent in the domain The proof of Theorem 39 relies on the same method as the proof of Theorem 38. For each $\alpha\in(1,4]$ a definitive result was obtained in [58]: a sharp domain of univalence for the class $\mathscr B_{\alpha}[0,1]$ was found there. The question of the existence and size of a domain of univalence for the classes $\mathscr B_{\alpha}[0,1]$, where $\alpha>4$, is still open. Theorem 40 (Solodov [58]). Let $f\in \mathscr B_{\alpha}[0,1]$, $\alpha\in (1,4]$. Then $f$ is univalent in the domain For any domain $\mathscr{U}$ such that $\mathscr D(\alpha)\subset\mathscr{U}\subset\mathbb D$ and $\mathscr U\ne \mathscr D(\alpha)$ there exists a function $f\in \mathscr B_{\alpha}[0,1]$ that is not univalent in $\mathscr{U}$. In Fig. 1 the reader can see the structure and mutual position of the domains of univalence from Theorems 37–40. The proof of Theorem 40 is based on the following inequality, which refines Lemma 6.1 for the class $\mathscr B[0,1]$. Lemma 6.2 (Solodov [58]). Let $f\in \mathscr B[0,1]$, and let $a,b\in \mathbb D$, $a\ne b$, be points such that $f(a)=f(b)=c$. Then where $\lambda(z)=-z\,{(1-\overline{z})}/{(1-z)}$. Remark 6.3. Since $|\lambda (z)|=|z|$ for all $z\in\mathbb D$, by Lemma 4.1 we have Proof of Lemma 6.2. Let $f\in \mathscr B[0,1]$. Then the function
$$
\begin{equation*}
g(z)=\frac{1-\overline{c}}{1-c}\,\frac{f(z)-c}{1-\overline{c}f(z)}
\end{equation*}
\notag
$$
belongs to $\mathscr B\{1\}$ and $g(a)=g(b)=0$. Moreover, $g(0)=\lambda(c)$ and its angular derivative at $z=1$ has the form By the Schwarz–Pick lemma $g$ admits a representation
$$
\begin{equation}
g(z)=\frac{1-\overline{a}}{1-a}\,\frac{z-a}{1-\overline{a}z}\, \frac{1-\overline{b}}{1-b}\,\frac{z-b}{1-\overline{b}z}\, h(z),
\end{equation}
\tag{6.9}
$$
where either $h\in \mathscr B\{1\}$ or $h(z)\equiv 1$. If $h(z)\equiv 1$, then equality holds in (6.7). Let $h\in \mathscr B\{1\}$. Then we see from (6.9) that $g(0)=\lambda(a)\lambda(b)h(0)$. Therefore, From (6.8) and (6.9) we can find the value of the angular derivative of $h$ at $z=1$:
$$
\begin{equation}
h'(1)=f'(1)\,\frac{1-|c|^2}{|1-c|^2}-\frac{1-|a|^2}{|1-a|^2}- \frac{1-|b|^2}{|1-b|^2}\,.
\end{equation}
\tag{6.11}
$$
Since $h\in\mathscr B\{1\}$, by the Julia–Carathéodory theorem we have Taking (6.10)–(6.12) into account we arrive at (6.7). $\Box$ A generalization of (6.7) to the case of $n$ distinct points was obtained in [50]. Proof of Theorem 40. Fix $\alpha\in (1,4]$. Suppose that there is a function $f$ in the class $\mathscr B_{\alpha}[0, 1]$ that is not univalent in $\mathscr D(\alpha)$, that is, there exist distinct points $a,b\in \mathscr D(\alpha)$ such that $f(a)=f(b)=c$, $c\in \mathbb D$. Then we have (6.7) by Lemma 6.2. However, using the properties of the involution $\lambda$ and overcoming some technical problems we can show that, in fact, we have the inequality reverse to (6.7). This contradiciton proves the first part of the theorem.
To prove the second part, for each point on the boundary of $\mathscr D(\alpha)$ it suffices to produce a function in $\mathscr B_{\alpha}[0,1]$ whose derivative vanishes at this point. For each $\theta\in(-\pi,\pi)$ consider the function
$$
\begin{equation*}
f_{\theta}(z)=z\frac{1-(\alpha-1)e^{i\theta}+\alpha e^{i\theta}z} {\alpha-(\alpha-1-e^{i\theta})z}\,.
\end{equation*}
\notag
$$
One readily verifies that for $\theta\in(-\pi,\pi)$ $f_{\theta}$ belongs to the class $\mathscr B_{\alpha}[0,1]$ and its derivative vanishes at the point
$$
\begin{equation}
z_{\alpha}(\theta)= \frac{\sqrt{\alpha-1}-e^{-i\theta/2}}{\sqrt{\alpha-1}+e^{i \theta/2}}\,.
\end{equation}
\tag{6.13}
$$
It is easy to see that formula (6.13) describes the curve bounding the domain $\mathscr D(\alpha)$. $\Box$ The shape of the domain of univalence for the class $\mathscr B_{\alpha}[0,1]$ (see (6.6)) is a key to finding a univalent covering domain for this class. Of course, since $\mathscr B_{\alpha}[0,1]$, $\alpha\in(1,2)$, embeds in $\mathscr B_{\alpha/(2-\alpha)}[0]$, all functions in $\mathscr{B}_\alpha[0,1]$ are invertible in some disc. In fact, we can find a much wider domain in place of this disc, which contains the interior fixed point and adjoins the boundary fixed point. The following definitive result holds. Theorem 41 (Kudryavtseva and Solodov [59]). Let $f\in \mathscr B_{\alpha}[0,1]$, where $\alpha\in(1,2)$. Then ther exists an inverse function of $f$ that maps the domain conformally onto some domain $\mathscr X\subset \mathbb D$. For any domain $\mathscr V$ such that $\mathscr W(\alpha)\subset \mathscr V \subset \mathbb D$ and $\mathscr V\ne \mathscr W(\alpha)$ there exists a function $f\in \mathscr B_{\alpha}[0, 1]$ that has no inverse in $\mathscr V$. Finally, for $\alpha\geqslant 2$ there exist no non-empty univalent covering domains for the class $\mathscr B_{\alpha}[0,1]$. Proof. To prove the first assertion it suffices to verify that for each $w\in\mathscr W(\alpha)$ the equation $f(z)=w$ has a unique solution in some subdomain of $\mathbb D$. We show that we can take $\mathscr D(\alpha)$ (see (6.6)) as this subdomain. For $z$ on the piecewise smooth curve
$$
\begin{equation*}
z=\frac{\sqrt{\alpha-1}-e^{-i\varphi/2}}{\sqrt{\alpha-1}+e^{i\varphi/2}}\,, \qquad \varphi\in[-\pi,\pi],
\end{equation*}
\notag
$$
which bounds the domain $\mathscr D(\alpha)$, the value $f(z)$ cannot occur in the interior of $\mathscr W(\alpha)$. In fact, by Schwarz’s lemma the function $g(z)=f(z)/z$ belongs to the class $\mathscr B$. Moreover, $g\in \mathscr B_{\alpha-1}\{1\}$. By the Julia–Carathéodory theorem the value of $g$ at $z$ satisfies
$$
\begin{equation*}
\frac{|1-g(z)|^2}{1-|g(z)|^2}\leqslant(\alpha-1)\,\frac{|1-z|^2}{1-|z|^2}\,.
\end{equation*}
\notag
$$
Hence $f(z)$ lies in the open disc with boundary
$$
\begin{equation}
\frac{|z-\zeta|^2}{|z|^2-|\zeta|^2}=(\alpha-1)\,\frac{|1-z|^2}{1-|z|^2}\,.
\end{equation}
\tag{6.14}
$$
By demonstrating that the boundary $\mathscr W(\alpha)$ has only one common point with the circle (6.14) we verify that $f(z)$ cannot occur in the interior of $\mathscr W(\alpha)$. Thus, as $z$ makes a turn along the boundary of $\mathscr D(\alpha)$, the argument of $f(z)-w$ increases by $2\pi$ for any $w$ in $\mathscr W(\alpha)$. Hence the equation $f(z)=w$ has a unique solution in $\mathscr D(\alpha)$.
We go over to the second part of the theorem. For each $\theta\in (-\pi,\pi)$ we look at the function
$$
\begin{equation*}
f_{\theta}(z)=z\frac{1-(\alpha-1)e^{i\theta}+\alpha e^{i\theta}z} {\alpha-(\alpha-1-e^{i\theta})z}\,.
\end{equation*}
\notag
$$
It belongs to the class $\mathscr B_{\alpha}[0,1]$, and its derivative vanishes at the point Hence the inverse function of $f_\theta$ has a branch point at Thus, each boundary point of $\mathscr W(\alpha)$ is an obstruction to extending the univalent covering domain. $\Box$ In the rest of this section we treat the case when the additional fixed point lies on the boundary. In a similar way we can assume without loss of generality that the fixed point is $z=-1$. The following problem arises: find a domain of univalence for the class
$$
\begin{equation*}
\mathscr B_{\alpha}[-1,1]=\mathscr B_{\alpha}\{1\}\cap \mathscr B[-1],
\end{equation*}
\notag
$$
for each $\alpha>1$ (we let $\mathscr B[-1]$ denote the class of functions $f\in\mathscr B$ such that $z=-1$ is the Denjoy–Wolff point). This problem has not been fully solved so far. On the other hand, for some values of $\alpha$ the existence of domains of univalence for the class $\mathscr B_{\alpha}[-1,1]$ follows from [60], where the symmetric class
$$
\begin{equation*}
\mathscr B_\alpha\{-1,1\}=\Bigl\{f\in \mathscr B\colon\angle\lim_{z\to\pm1} f(z)=f(\pm 1)=\pm 1,\, f'(1)f'(-1)\leqslant \alpha\Bigr\}
\end{equation*}
\notag
$$
was considered. Such a class is a quite natural object of investigation because the functional $f'(1)f'(-1)$ is bounded below by 1 by the Julia–Carathéodory theorem. On the other hand, since $\mathscr B_\alpha[-1,1]\subset\mathscr B_\alpha\{-1,1\}$, we can conclude about the existence of a univalence domain for $\mathscr B_\alpha[-1,1]$ from the existence of such a domain for the wider class $\mathscr B_\alpha\{-1,1\}$. The following result was obtained in [60]. Theorem 42 (Goryainov [60]). Let $f\in \mathscr B_{\alpha}\{-1,1\}$, where $\alpha\in(1,9)$. Then $f$ is univalent in the domain where $y^*(\alpha)$ is the unique root of the equation in the interval $0<y<\pi/2$. Remark 6.4. The domain (6.15) can also be described in an equivalent way: Proof of Theorem 42. Let $f\in \mathscr B_{\alpha}\{-1,1\}$, where $\alpha\in (1,9)$. By Lemma 5.2 the function $h(z)=(1+f(z))/(1-f(z))$ has a representation of the form (5.3). Then $g(\zeta)=h\circ L(\zeta)$, where $L(\zeta)=(\zeta-1)/(\zeta+1)$, has the following form:
$$
\begin{equation*}
g(\zeta)=\rho\biggl(\frac{\zeta}{\alpha}+\biggl(1-\frac{1}{\alpha}\biggr) \zeta\int_{\mathbb T}\frac{1+\varkappa}{(1-\varkappa)\zeta +1+\varkappa}\, d\mu(\varkappa)\biggr).
\end{equation*}
\notag
$$
We can find a lower estimate for the real part of the derivative of $g$:
$$
\begin{equation*}
\operatorname{Re}g'(\zeta)\geqslant \rho\biggl(\frac{1}{\alpha}+ \biggl(1-\frac{1}{\alpha}\biggr)\min_{\varkappa\in \mathbb T}\operatorname{Re} \frac{(1+\varkappa)^2}{((1-\varkappa)\zeta +1+\varkappa))^2}\biggr),
\end{equation*}
\notag
$$
which shows that $\operatorname{Re} g'(\zeta)>0$ at each point $\zeta$ in the convex domain
$$
\begin{equation*}
\mathscr C(\alpha)=\{(x,y)\in \mathbb R^2 \colon |y|<\sqrt{m(\alpha)}\,x\}.
\end{equation*}
\notag
$$
By the Noshiro–Warshawski theorem (Theorem 9) the function $g$ is univalent in $\mathscr C(\alpha)$. However, then $f=L\circ g\circ L^{-1}$ is univalent in the domain $L(\mathscr C(\alpha))$, which coincides with $\mathscr J(\alpha)$. $\Box$ For each $\alpha\in(1,9)$ the domain $\mathscr J(\alpha)$ contains the real diameter and is bounded by two arcs of circles passing through the points $z=\pm1$. It is easy to see that the condition $\alpha<9$ is sharp. In fact, $f(z)=z^3$ belongs to the class $\mathscr B_9\{-1,1\}$ but is not univalent in any neighbourhood of the real diameter. The domains of univalence $\mathscr J(\alpha)$ are not best possible. In [52] an upper bound was obtained for each $\alpha\in(1,9)$: a domain was constructed such that a domain of univalence for the whole class $\mathscr B_{\alpha}\{-1,1\}$ cannot be extended over its boundary. However, this upper estimate, which supplements Theorem 42, does not answer the question of whether or not the domains $\mathscr J(\alpha)$ are best possible. On the other hand this upper estimate is quite effective for the subclass $\mathscr D_\alpha\{-1,1\}$ of functions preserving the real diameter:
$$
\begin{equation*}
\mathscr D_{\alpha}\{-1,1\}=\{f\in \mathscr B_{\alpha}\{-1,1\}\colon f((-1,1))=(-1,1)\}, \qquad \alpha>1,
\end{equation*}
\notag
$$
which is the most interesting class from the standpoint of applications. Theorem 43 (Kudryavtseva and Solodov [52]). Let $f\in \mathscr D_{\alpha}\{-1,1\}$, where $\alpha\in(1,9)$. Then $f$ is univalent in the domain where Theorem 44 (Kudryavtseva and Solodov [52]). Let $\alpha>1$. Then for each boundary point $z_0$ of the domain there exists a function $f_{z_0}\in \mathscr D_{\alpha}\{-1,1\}$ whose derivative vanishes at $z_0$. Proof of Theorem 43. Let $f\in \mathscr D_{\alpha}\{-1,1\}$, where $\alpha\in(1,9)$. By Lemma 5.3 the function $h(z)=(1+f(z))/(1-f(z))$ has a representation of the form (5.6). Then $g(\zeta)=h\circ L(\zeta)$, where $L(\zeta)=(\zeta-1)/(\zeta+1)$, has the following form:
$$
\begin{equation*}
g(\zeta)=\rho\biggl(\frac\zeta\alpha+\biggl(1-\frac{1}\alpha\biggr)\zeta \int_{\mathbb T}\frac{1+\operatorname{Re}\varkappa}{1+\zeta^2+ \operatorname{Re}\varkappa\,(1-\zeta^2)}\,d\mu(\varkappa)\biggr).
\end{equation*}
\notag
$$
We can estimate the real part of the derivative of $g$ from below:
$$
\begin{equation*}
\operatorname{Re} g'(\zeta) \geqslant \rho\biggl(\frac{1}\alpha+ \biggl(1-\frac{1}\alpha\biggr)\min_{\varkappa\in \mathbb T}\operatorname{Re} \frac{(1+\operatorname{Re}\varkappa) \bigl(1-\zeta^2+\operatorname{Re}\varkappa\,(1+\zeta^2)\bigr)} {\bigl(1+\zeta^2+\operatorname{Re}\varkappa\,(1-\zeta^2)\bigr)^2}\biggr).
\end{equation*}
\notag
$$
Then we can find a domain such that $\operatorname{Re}g'(\zeta)>0$ in it. By the Noshiro–Warshawski theorem (Theorem 9) $g$ is univalent in this domain. Thus, the function $f=L\circ g\circ L^{-1}$ is univalent in the $L$-image of this domain, which coincides with $\underline{\mathscr U}(\alpha)$. $\Box$ Proof of Theorem 44. For each $s\in(-1,1)$ let where
The function $f_s$ can be represented as the composition $f_s(z)=T_s^{-1}\circ f_0\circ T_s(z)$, where
$$
\begin{equation*}
T_s(z)=\frac{z-s}{1-s z} \quad\text{and}\quad f_0(z)=z\biggl(z^2-\frac{\sqrt{\alpha}-3}{\sqrt{\alpha}+1}\biggr) \biggl(1-\frac{\sqrt{\alpha}-3}{\sqrt{\alpha}+1}\,z^2\biggr)^{-1}.
\end{equation*}
\notag
$$
Since $s \in (-1,1)$, $T_s$ is a linear fractional transformation of the disc $\mathbb D$, and since $\alpha\in(1,9)$, the function $f_0$ maps $\mathbb D$ into itself. Thus, the composition $f_s$ belongs to $\mathscr B$.
We can easily verify that for $s\in(-1,1)$ the function $f_s$ preserves the real diameter and, as $f'_{s}(\pm 1)=\sqrt{\alpha}$ , it follows that $f_s\in \mathscr D_{\alpha}\{-1,1\}$. On the other hand direct calculations show that the derivative of $f_s$ vanishes at the points
$$
\begin{equation*}
z_{1,2}(s)=\frac{-2\sqrt[4]{\alpha}+p_1(\alpha)\pm ip_2(\alpha)s} {(-2\sqrt[4]{\alpha}+p_1(\alpha))s\pm ip_2(\alpha)}\,,
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
p_1(\alpha)=\sqrt{(\sqrt{\alpha}-1)(3+\sqrt{\alpha}\,)}\,,\qquad p_2(\alpha)=\sqrt{(\sqrt{\alpha}+1)(3-\sqrt{\alpha}\,)}\,.
\end{equation*}
\notag
$$
The curves $z_1(s)$ and $z_2(s)$, $s\in (-1,1)$, are the arcs bounding the domain $\overline{\mathscr U}(\alpha)$ (see (6.16)). $\Box$ Since boundary points of $\overline{\mathscr U} (\alpha)$ are zeros of the derivatives of functions in $\mathscr D_{\alpha}\{-1,1\}$, each point on the boundary of $\overline{\mathscr U} (\alpha)$ is an obstacle to extending univalence. Thus, Theorem 44 gives an upper bound for domains of univalence of the class $\mathscr D_{\alpha}\{-1, 1\}$, and therefore also of $\mathscr B_{\alpha}\{-1,1\}$. Now we examine the difference between the upper and lower bounds in a neighbourhood of the extreme values of $\alpha$ (see Fig. 2). As the domains ${\mathscr J}(\alpha)$, $\underline{\mathscr U} (\alpha)$, and $\overline{\mathscr U}(\alpha)$ are bounded by arcs of circles, it is sufficient to compare the tangent functions of the angles at $z=\pm 1$ between these arcs and the real axis. Let $\varphi_1(\alpha)$ and $\varphi_2(\alpha)$ be the angles for the domains $\mathscr{J}(\alpha)$ and $\mathscr{\overline U}(\alpha)$, respectively. Then we have the following asymptotic relations:
$$
\begin{equation*}
\begin{alignedat}{3} \tan \varphi_1(\alpha)&\sim\frac{1}{\sqrt{\alpha-1}}\,,&\qquad \tan\varphi_2(\alpha)&\sim\frac{\sqrt{2}}{\sqrt{\alpha-1}} &\qquad \text{as}\quad \alpha &\to 1+0; \\ \tan\varphi_1(\alpha)&\sim\frac{9-\alpha}{16\sqrt{3}}\,,&\qquad \tan\varphi_2(\alpha)&\sim\frac{\sqrt{9-\alpha}}{3\sqrt{2}} &\qquad \text{as}\quad \alpha &\to 9-0. \end{alignedat}
\end{equation*}
\notag
$$
For $\alpha$ close to 1 the upper and lower bounds have the same order, but for $\alpha$ close to $9$ there is a large gap between the two bounds. As regards the lower bound for the class $\mathscr D_{\alpha}\{-1,1\}$, it does not just produce a larger domain (as expected), but it is asymptotically close to the upper bound, both as $\alpha\to 1+0$ and as $\alpha\to 9-0$. Let $\varphi_3(\alpha)$ denote the angle at the points $z=\pm 1$ between the boundary of $\mathscr{\underline U}(\alpha)$ and the real axis. Then
$$
\begin{equation*}
\begin{alignedat}{2} \tan \varphi_2(\alpha) &\sim \tan\varphi_3(\alpha)\sim \frac{\sqrt{2}}{\sqrt{\alpha-1}}&\qquad \text{as}\quad \alpha &\to 1+0; \\ \tan\varphi_2(\alpha) &\sim \tan\varphi_3(\alpha)\sim \frac{\sqrt{9-\alpha}}{3\sqrt{2}} &\qquad \text{as}\quad \alpha &\to 9-0. \end{alignedat}
\end{equation*}
\notag
$$
7. One-parameter semigroups and infinitesimal generatorsIf the domain of definition of a function $f$ agrees with its range, then the non-negative integer iterates of $f$ are well defined: $f^0(z)\equiv z$, $f^{1}(z)=f(z)$, and $f^{n}(z)=f\circ f^{n-1}(z)$ for $n=2,3,\dots$ . One of the central questions here is the existence of fractional iterates, that is, a family of functions $\{f^t\}_{t\geqslant 0}$, such that $f^0(z)\equiv z$, $f^{1}(z)=f(z)$, and $f^{s+t}(z)=f^s\circ f^{t}(z)$ for $s,t\geqslant 0$. In this topic one distinguishes three quite different cases. The first investigations related to the local case, when both $f$ and its iterates are defined by power series in a neighbourhood (depending on the iterate) of their common fixed point $z_0$, $f(z_0)=z_0$. Schröder [61] related the problem of fractional iterates with a solution of a functional equation, while Koenigs [62] introduced a construction solving this equation and showed that only slight restrictions should be imposed on the function $f$ in order that fractional iterates exist, namely, $|f'(z_0)|\ne 0$ and $|f'(z_0)|\ne 1$. Subsequently, the problem of fractional iterates was considered for meromorphic functions, when the whole plane except the set of isolated singular points, poles, is the domain of definition. The results obtained there are polar opposite to the local case. For instance, Baker [63], and Karlin and McGregor [64] showed that fractional iterates only exist for linear fractional functions. The third case concerns holomorphic functions in some domain that take values in this domain again. This is the most diversified case from the standpoint of both results and applications. The most significant step in investigations was regarding the family of fractional iterates as a one-parameter semigroup. Turning to a rigorous presentation, we introduce the requisite notation and definitions. The set $\mathscr B$ of holomorphic self-maps of the unit disc $\mathbb D$ is a topological semigroup with respect to the operation of composition and the topology of locally uniform convergence in ${\mathbb D}$, the identity map $f(z)\equiv z$ being the identity element. We regard $\mathbb R_+=\{ t\in {\mathbb R}\colon t\geqslant 0 \}$ as an additive semigroup with the usual topology of real numbers. By a one-parameter semigroup of $\mathscr B$ we mean a continuous homomorphism $t\mapsto f^t$ from the semigroup $\mathbb R_+$ to $\mathscr B$. That is, the family $\{f^t\}_{t\geqslant 0}$ of functions $\mathscr B$ must satisfy the following conditions:
Since for non-negative integers $t$ the elements $f^t$ of the one-parameter semigroup are integer iterates of $f=f^1$, for $t\geqslant 0$ the maps $f^t$ are called fractional iterates of $f$. We say that $f \in \mathscr B$ is embeddable in a one-parameter semigroup of $\mathscr B$ if there exists a one-parameter semigroup $t\mapsto f^t$ of $\mathscr B$ such that $f^1=f$. Thus the problem of fractional iterates is equivalent to the embeddability of the function $f$ in a one-parameter semigroup $t\mapsto f^t$ of $\mathscr B$. Regarding a one-parameter semigroup as a dynamical system we arrive in a natural way to the notion of infinitesimal generator. Note that every one-parameter semigroup $t\mapsto f^t$ of $\mathscr B$ is differentiable with respect to $t$ [65]; moreover, it is infinitely differentiable [66]. The derivative
$$
\begin{equation*}
\frac{\partial}{\partial t}f^t(z)\bigg|_{t=0}= \lim_{t\to + 0}\frac{f^t(z)-z}{t}=v(z)
\end{equation*}
\notag
$$
is an analytic function in the disc ${\mathbb D}$; it specifies the one-parameter semigroup $t\mapsto f^t$ by means of the differential equation with initial condition $f^t(z)\big|_{t=0}=z$. The function $v$ is called the infinitesimal generator of the one-parameter semigroup $t\mapsto f^t$. The investigation of the form of the infinitesimal generator is closely connected with an analysis of fixed points. Among the common fixed points of the functions $f^t$, $t\geqslant 0$, the so-called Denjoy–Wolff point $q$ stands out, which is an attractingg point. Theorem 45 (Denjoy and Wolff [67]–[69]). Let $f \in \mathscr B$ be a function distinct from a linear fractional self-map of ${\mathbb D}$. Then there exists a unique point $q\in \overline{\mathbb D}$ such that the sequence of integer iterates $f^{n}=f\circ f^{n-1}$, $n=2,3,\dots$ , of $f=f^1$ converges to $q$ locally uniformly in ${\mathbb D}$. In addition, if $ q\in \mathbb T$, then the angular limits exist and the relations $f(q)=q$ and $0<f'(q)\leqslant 1$ hold. The infinitesimal generator of a semigroup $t\mapsto f^t$ of $\mathscr B$ was explicitly found by Löwner [51] in the case when the Denjoy–Wolff point lies in the interior of $ \mathbb D$ and by Berkson and Porta [65] in the case when this is a boundary point. We present a general expression for the infinitesimal generators of one-parameter semigroups $t\mapsto f^t$ of $\mathscr B$ in terms of the Denjoy–Wolff point. Let $\mathscr B[q]$, $q\in \overline{\mathbb D}$, denote the set of functions $f$ in $\mathscr B$ such that $q$ is their Denjoy–Wolff point. We also assume that the identity map $f(z)\equiv z$ belongs to all sets $\mathscr B[q]$. Then $\mathscr B[q]$ is a subsemigroup with identity element of the semigroup $\mathscr B$. Since $\mathscr B[q]$ and $\mathscr B[\widetilde{q}]$ have no common elements but the identity for $q\ne \widetilde{q}$, each one-parameter semigroup $t\mapsto f^t$ in $\mathscr B$ is a subsemigroup of some $\mathscr B[q]$, where $q\in \overline{\mathbb D}$. The following result, which is called the Berkson–Porta formula, describes the form of the infinitesimal generators of one-parameter semigroups in $\mathscr B[q]$. Theorem 46 (Berkson–Porta formula). A holomorphic function $v$ in $\mathbb D$ is the infinitesimal generator of a one-parameter semigroup $t\mapsto f^t$ in $\mathscr B[q]$ for some $q\in \overline{\mathbb D}$ if and only if it has a representation where $p$ is a holomorphic function with non-negative real part in ${\mathbb D}$. Proof. Let $t\mapsto f^t$ be a one-parameter semigroup in $\mathscr B[0]$ with Denjoy–Wolff point $q=0$ and infinitesimal generator $v$. We show that $v$ has a representation (7.1). By Schwarz’s lemma $|f^t(z)|\leqslant |z|$ for all $t\geqslant 0$ and $z\in\mathbb D$. Hence, for an appropriate choice of the branch of the argument we have which can also be written as Hence $\operatorname{Re}(-v(z)/z)\geqslant 0$. Thus, $v(z)=-z p(z)$, where $p$ is a holomorphic function with non-negative real part in $ {\mathbb D}$. The case when $p$ is identically equal to an imaginary constant corresponds to the one-parameter rotation semigroup of the unit disc $\mathbb D$.
Conversely, let us show that if $v(z)=-zp(z)$, where $p$ is a holomorphic function with non-negative real part in ${\mathbb D}$, then there exists a one-parameter semigroup in $\mathscr B[0]$ with this infinitesimal generator. To do this consider the function
$$
\begin{equation*}
F(z)=z\exp\biggl\{\int_0^z \frac{p(0)-p(\zeta)}{\zeta p(\zeta)}\,d\zeta\biggr\}.
\end{equation*}
\notag
$$
Since by Theorem 14 the function $F$ is spiral-like, that is, it is univalent in $\mathbb D$ and takes it to a domain that, together with each point $F(z)$, $z\in \mathbb D$, contains also the spiral $w(t)=F(z)e^{-p(0)t}$, $t\geqslant 0$. Hence we can define the one-parameter semigroup $t\mapsto f^t$ of $\mathscr B$ by setting where $t\geqslant 0$. Since $F(0)=0$, the point $z=0$ is fixed by all functions $f^t$, $t>0$. Moreover, the infinitesimal generator $v$ of the one-parameter semigroup $t\mapsto f^t$ has the following form:
$$
\begin{equation*}
v(z)=\frac{\partial}{\partial t}f^t(z)\bigg|_{t=0}= -\frac{p(0)F(z)}{F'(z)}=-z p(z).
\end{equation*}
\notag
$$
Now we extend this result to the case when the Denjoy–Wolff point $q$ has an arbitrary position in the interior of $\mathbb D$. Consider the linear fractional transformation $L(z)={(z-q)}/{(1-\overline{q}z)}$, which takes the disc ${\mathbb D}$ to itself and satisfies $L(q)=0$. If $t\mapsto f^t$ is a one-parameter semigroup in $\mathscr B[0]$ with infinitesimal generator $v$, then $t\mapsto \widetilde{f}^t$, where $\widetilde{f}^t(z)=L^{-1}\circ f^t \circ L(z)$ for $t\geqslant 0$, is a one-parameter semigroup in $\mathscr B[q]$, $q\in \mathbb D$, with infinitesimal generator
$$
\begin{equation*}
\widetilde{v}(z)=\frac{v(L(z))}{L'(z)}=(q-z)(1-\overline{q}z)p(z),
\end{equation*}
\notag
$$
where $p$ is a holomorphic function with non-negative real part in ${\mathbb D}$.
We turn to the case of Denjoy–Wolff point on the boundary. Let $t\mapsto f^t$ be a one-parameter semigroup in $\mathscr B[1]$ with infinitesimal generator $v$. By the Julia–Carathéodory theorem $f^t$ transforms the horocycle at $1$ with parameter $k>0$,
$$
\begin{equation*}
\mathbb H_k=\biggl\{z\in\mathbb D\colon \frac{|1-z|^2}{1-|z|^2}< k\biggr\},
\end{equation*}
\notag
$$
into itself. Hence the vector $f^t(z)-z$ points inward the horocycle $\mathbb H_k$, which contains $z$ on its boundary. In view of the relation between the central and inscribed angles resting on the same arc, we conclude that the inward normal at $z$ to the boundary of $\mathbb H_k$ is co-directed with the vector $(1-z)^2$. Therefore,
$$
\begin{equation*}
\biggl|\arg \frac{f^t(z)-z}{(1-z)^2}\biggr|\leqslant \frac{\pi}{2}\,,
\end{equation*}
\notag
$$
or
$$
\begin{equation*}
\operatorname{Re}\frac{f^t(z)-z}{(1-z)^2}\geqslant 0,
\end{equation*}
\notag
$$
for all $t\geqslant 0$ and $z\in\mathbb D$. Hence $\operatorname{Re}\bigl(v(z)/(1-z)^2\bigr)\geqslant 0$. Thus, $v(z)=(1-z)^2 p(z)$, where $p$ is a holomorphic function with positive real part in ${\mathbb D}$. The case when $p$ is identically equal to an imaginary constant corresponds to a one-parameter semigroup of linear fractional transformations of $\mathbb D$ into itself.
Conversely, let $p$ be a holomorphic function in ${\mathbb D}$ such that $\operatorname{Re} p(z)>0$ for $z\in \mathbb D$. Consider the function Since where $G(z)=1/(1-z)$, $F$ is a close-to-convex function. To understand the geometry of the image $F(\mathbb D)$ we study the properties of the $F$-images of the horocycles $\mathbb H_k$, $k>0$.Let $z=z(s)$, $-\infty<s<\infty$, be a parametrization of the boundary of $\mathbb H_k$ inducing the positive orientation. Then it follows from the above observation on a normal to the boundary of a horocycle that $z'(s)/(1-z(s))^2$ lies on the negative part of the imaginary axis. However, then
$$
\begin{equation*}
\frac{d}{ds}\operatorname{Im}F(z(s))= \operatorname{Im}\bigl(F'(z(s))z'(s)\bigr)= \operatorname{Im}\frac{z'(s)}{(1-z(s))^2\,p(z(s))}<0.
\end{equation*}
\notag
$$
This means that the image $F(\mathbb H_k)$ of $\mathbb H_k$ is a domain that, together with any point $F(z)$, also contains the ray $w(t)=F(z)+t$, $t\geqslant 0$. This property extends to the whole of the image $F(\mathbb D)$ of the unit disc $\mathbb D$. Hence for each $t\geqslant 0$ we have a well-defined function It is easy to see that $t\mapsto f^t$ is a one-parameter semigroup in $\mathscr B[1]$ with infinitesimal generator
$$
\begin{equation*}
v(z)=\frac{\partial}{\partial t}f^t(z)\bigg|_{t=0}= \frac{1}{F'(z)}=(1-z)^2p(z).
\end{equation*}
\notag
$$
To extend this result to the case when the Denjoy–Wolff point $q$ has an arbitrary position on the boundary of $\mathbb D$ it is sufficient to note that if $t\mapsto f^t$ is the one-parameter semigroup in $\mathscr B[1]$ with infinitesimal generator $v$, then $t\mapsto \widetilde{f}^t$, where $\widetilde{f}^t(z)=q f^t(\overline{q}z)$, $t\geqslant 0$, is a one-parameter semigroup in $\mathscr B[q]$ with infinitesimal generator $\widetilde{v}(z)=qv(\overline{q}z)=(q-z)(1-\overline{q}z)p(z)$, where $p$ is a holomorphic function with non-negative real part in ${\mathbb D}$. $\Box$ Subsequently, there were investigations of how the Berkson–Porta formula transforms in the case when, apart from the Denjoy–Wolff point $q$, $|q|\leqslant 1$, the function $f\in \mathscr B$, also has other fixed points (as already mentioned, these points must lie on the unit circle $\mathbb T$). Various forms of this question were considered by many authors (see, for instance, [70]–[73]). In [74] an analogue of the Berkson–Porta formula for the infinitesimal generator of a one-parameter semigroup with two fixed points was considered. Before we formulate this result, we introduce our notation and present several auxiliary results. Let $\mathscr B[q;a]$ denote the subsemigroup of $\mathscr B[q]$ functions $f$ in which fix a point $a \in \mathbb T$, $a\ne q$, and have finite angular derivatives at this point. Consider the class $\mathscr{Q}$ of holomorphic functions $h$ in $\mathbb D$ that admit an integral representation of the form
$$
\begin{equation}
h(z)=\int_{\mathbb T}\frac{1-\varkappa}{1-\varkappa z}\,d\mu(\varkappa),
\end{equation}
\tag{7.2}
$$
where $\mu$ is a probability measure on $\mathbb T$. The following statements give two equivalent definitions of the class $\mathscr{Q}$ and describe some properties of the functions in this class. Lemma 7.1 (Goryainov and Kudryavtseva [74]). A holomorphic function $h$ in $\mathbb D$ belongs to the class $\mathscr{Q}$ if and only if it has a representation where $\varphi$ is a holomorphic function in $\mathbb D$ such that $|\varphi(z)|\leqslant 1$ for $z\in \mathbb D$. Proof. Let $h \in \mathscr{Q}$, and assume that the representation (7.2) with probability measure $\mu$ on $\mathbb T$ is valid. Note that
$$
\begin{equation*}
\frac{1-\varkappa}{1-\varkappa z}= \frac{1}{2}\biggl(\biggl(\frac{1+\varkappa z}{1-\varkappa z}+1\biggr)- \frac{1}{z}\biggl(\frac{1+\varkappa z}{1-\varkappa z}-1\biggr)\biggr).
\end{equation*}
\notag
$$
So we arrive at the equality
$$
\begin{equation}
h(z)=\frac{1}{2}\biggl(\bigl(p(z)+1\bigr)- \frac{1}{z}\bigl(p(z)-1\bigr)\biggr),
\end{equation}
\tag{7.4}
$$
where
$$
\begin{equation}
p(z)=\int_{\mathbb T}\frac{1+\varkappa z}{1-\varkappa z}\,d\mu(\varkappa).
\end{equation}
\tag{7.5}
$$
Since $p$ is holomorphic, has a positive real part in $\mathbb D$, and $p(0)=1$, the function satisfies the assumptions of Schwarz’s lemma, that is, $\psi(0)=0$ and $|\psi(z)|\leqslant 1$ for $z \in \mathbb D$. However, then $\varphi(z)=\psi(z)/z$ is a holomorphic function in $\mathbb D$ such that $|\varphi(z)|\leqslant 1$ for $z\in \mathbb D$. Expressing $p$ in terms of $\varphi$ and substituting this expression into (7.4) we arrive at (7.3).
Conversely, assume that $h$ has a representation (7.3) for some holomorphic function $\varphi$ in $\mathbb D$ such that $|\varphi(z)|\leqslant 1$ for $z\in \mathbb D$. Setting we obtain a holomorphic function with positive real part in $\mathbb D$ such that $p(0)=1$. By the Riesz–Herglotz theorem there exists a probability measure $\mu$ on $\mathbb T$ that is connected with $p$ by (7.5). However, then
$$
\begin{equation*}
\begin{aligned} \, h(z)&=\frac{1-\varphi(z)}{1-z\varphi(z)}= \frac{1}{2}\biggl(\biggl(\frac{1+z\varphi(z)}{1-z\varphi(z)}+1\biggr)- \frac{1}{z}\biggl(\frac{1+z\varphi(z)}{1-z\varphi(z)}-1\biggr)\biggr) \\ &=\frac{1}{2}\biggl((p(z)+1)- \frac{1}{z}(p(z)-1)\biggr)=\int_{\mathbb T} \frac{1-\varkappa}{1-\varkappa z}\,d\mu(\varkappa). \qquad\square \end{aligned}
\end{equation*}
\notag
$$
Remark 7.1. It follows from (7.3) that a function $h$ in $\mathscr{Q}$ is either identically equal to zero, or $h(z)\ne 0$ for $z\in \mathbb D$. Lemma 7.2 (Goryainov and Kudryavtseva [74]). Let $h$ be a holomorphic function in $\mathbb D$ such that $h(z)\not\equiv 0$. Then $h$ belongs to the class $\mathscr{Q}$ if and only if the following representation holds: where $g$ is a holomorphic functions with non-negative real part in $\mathbb D$. Proof. Let $h \in \mathscr{Q}$ and assume that $h(z)\not\equiv 0$. By Lemma 7.1 the function $h$ has a representation (7.3) for some holomorphic function $\varphi$ in $\mathbb D$ such that $|\varphi(z)|\leqslant 1$ for $z\in \mathbb D$ and $\varphi(z) \not\equiv 1$. Therefore,
$$
\begin{equation*}
\begin{aligned} \, \frac{1}{h(z)}&=\frac{1-z\varphi(z)}{1-\varphi(z)}= \frac{1}{1-\varphi(z)}-z\frac{\varphi(z)}{1-\varphi(z)} \\ &=\frac{1}{2}\biggl(\biggl(\frac{1+\varphi(z)}{1-\varphi(z)}+1\biggr)- z\biggl(\frac{1+\varphi(z)}{1-\varphi(z)}-1\biggr)\biggr)= \frac{1}{2}(1-z)\biggl(\frac{1+z}{1-z}+g(z)\biggr), \end{aligned}
\end{equation*}
\notag
$$
where $g(z)=(1+\varphi(z))/(1-\varphi(z))$.
Conversely, assume that the representation (7.6) holds; then $\varphi(z)=(g(z)-1)/(g(z)+1)$ is a holomorphic function in $\mathbb D$ and $|\varphi(z)|\leqslant 1$ for $z\in \mathbb D$. In this case $g(z)=(1+\varphi(z))/(1-\varphi(z))$ and
$$
\begin{equation*}
h(z)=2\biggl[(1-z){\biggl(\frac{1+z}{1-z}+ \frac{1+\varphi(z)}{1-\varphi(z)}\biggr)}\biggr]^{-1}= \frac{1-\varphi(z)}{1-z\varphi(z)}\,,
\end{equation*}
\notag
$$
that is, $h$ is a function in $\mathscr{Q}$ by Lemma 7.1. $\Box$ Lemma 7.3 (Goryainov and Kudryavtseva [74]). Let $z_0$ be a point in the unit disc $\mathbb D$. Then the set is the closed disc Moreover, the boundary point $\zeta \in \partial K_{z_0}$ is the contribution of a unique function in $\mathscr{Q}$, which is defined by the equality Proof. It follows from the definition (7.2) of the class $\mathscr{Q}$ that $K_{z_0}$ is the closed convex hull of the curve $\Gamma_{z_0}$, the image of the unit circle $\mathbb{T}$ under the linear fractional transformation
By the circle property of linear fractional transformations, $\Gamma_{z_0}$ itself is a circle. By the symmetry principle its centre $w_0$ is the image of the point $\overline{z}_0$ symmetric to the pole $1/z_0$ of $L$ with respect to the unit circle $\mathbb{T}$. Therefore,
$$
\begin{equation*}
w_0=L(\overline{z}_0)=\frac{1-\overline{z}_0}{1-|z_0|^2}\,.
\end{equation*}
\notag
$$
Also note that $\Gamma_{z_0}$ contains the origin. Hence this circle has radius $|w_0|$.
A boundary point $\zeta$ of $K_{z_0}$ is the contribution of the function $h$ corresponding, via the representation (7.2), to the Dirac measure $\mu$ concentrated at the point $\varkappa \in \mathbb{T}$ such that $\zeta=L(\varkappa)$. $\Box$ Theorem 47 (Goryainov and Kudryavtseva [74]). A holomorphic function $v$ in $\mathbb D$ is the infinitesimal generator of a one-parameter semigroup in $\mathscr B[q;a]$, where $q\in \overline{\mathbb D}$ and $a\in \mathbb{T}$, if and only if it admits a representation for some $\alpha > 0$ and $h\in \mathscr{Q}$. Proof. Let $t\mapsto f^t$ be a non-trivial ($f^t(z)\not\equiv z$ for $t>0$) one-parameter semigroup in $\mathscr B[q;a]$ with infinitesimal generator $v$. We show that $v$ has a representation (7.8) for some function $h$ in the class $\mathscr{Q}$.
First let $q=0$ and $a=1$. For each function $f \in \mathscr B[0;1]$ we have $f'(1)\geqslant 1$, with equality only for $f(z)\equiv z$. Hence it follows from the relation $f^{t+s}(z)=f^t\circ f^s(z)$ and the properties of the angular derivative that $\beta(t)=(f^t)'(1)$ is an increasing function such that $\beta(t+s)=\beta(t)\beta(s)$ and $\beta(0)=1$. Therefore, $\beta(t)=e^{\lambda t}$, where $\lambda=\ln(f^1)'(1)>0$. Consider the family of linear fractional transformations
$$
\begin{equation*}
U_t(z)=\frac{z+u(t)}{1+u(t)z}, \quad t>0, \quad\text{where } u(t)=\frac{e^{\lambda t}-1}{e^{\lambda t}+1}\,.
\end{equation*}
\notag
$$
They map the unit disc ${\mathbb D}$ onto itself and fix the points $z=\pm 1$. Setting $g_t(z)=f^t\circ U_t(z)$ we obtain a family $\{g_t\}_{t\geqslant 0}$ of self-maps of ${\mathbb D}$, which is $t$-differentiable, and we have $g_0(z)\equiv z$. Moreover, for each $t>0$ the function $g_t$ fixes $z=1$ and has the angular derivative $(g_t)'(1)=1$ at this point. By the Julia–Carathéodory theorem the image $g_t(z)$ of a point $z\in \mathbb D$ lies in the interior of the horocycle containing $z$ at its boundary, that is, in the disc with centre and radius This means that the angle between the vectors $\xi-z$ and $g_t(z)-z$ is at most $\pi/2$. We also observe that
$$
\begin{equation*}
\begin{aligned} \, \arg\biggl\{\frac{1-|z|^2}{2(1-\operatorname{Re}z)}-z\biggr\}&= \arg\{1-|z|^2-2z(1-\operatorname{Re}z)\} \\ &=\arg\bigl\{1-z\overline{z}-z\bigl((1-z)+ (1-\overline{z})\bigr)\bigr\}=\arg\{(1-z)^2\}. \end{aligned}
\end{equation*}
\notag
$$
So we obtain for $z \in \mathbb D$. Hence
$$
\begin{equation*}
\frac{\partial}{\partial t}g_t(z)\bigg|_{t=0}= \lim_{t\to 0}\frac{g_t(z)-z}{t}=(1-z)^2p(z),
\end{equation*}
\notag
$$
where $p$ is a holomorphic function with non-negative real part in $\mathbb D$.
On the other hand
$$
\begin{equation*}
\frac{\partial}{\partial t}g_t(z)\bigg|_{t=0}= \frac{\partial}{\partial t}f^t\bigl(U_t(z)\bigr)\bigg|_{t=0}+ (f^t)'\bigl(U_t(z)\bigr)\frac{\partial}{\partial t}U_t(z)\bigg|_{t=0}= v(z)+\frac{\lambda}{2}(1-z^2),
\end{equation*}
\notag
$$
where $v$ is the infinitesimal generator of the one-parameter semigroup $t\mapsto f^t$.
Thus we arrive at the equality
$$
\begin{equation*}
v(z)=(1-z)\biggl((1-z)p(z)-\frac{\lambda}{2}\,(1+z)\biggr).
\end{equation*}
\notag
$$
By assumption $q=0$, that is, $z=0$ is a fixed point of the functions $f^t$, $t>0$, and therefore $v(0)=0$. However, then $p(0)=\lambda/2$, and $p_1(z)=(2/{\lambda})p(z)$ has a positive real part in $\mathbb D$ and is normalized by the condition $p_1(0)=1$. By the Riesz–Herglotz theorem there exists a probability measure $\mu$ on $\mathbb T$ such that
$$
\begin{equation*}
p_1(z)=\int_{\mathbb T}\frac{1+\varkappa z}{1-\varkappa z}\,d\mu(\varkappa).
\end{equation*}
\notag
$$
Then we can represent the infinitesimal generator $v$ in the following form:
$$
\begin{equation*}
\begin{aligned} \, v(z)&=\frac{\lambda}{2}\,(1-z)\biggl((1-z)\int_{\mathbb T} \frac{1+\varkappa z}{1-\varkappa z}\:d\mu(\varkappa)-(1+z)\biggr) \\ &=-\lambda z (1-z) \int_{\mathbb T} \frac{1-\varkappa}{1-\varkappa z}\,d\mu(\varkappa). \end{aligned}
\end{equation*}
\notag
$$
This is the representation from the statement of the theorem for $q=0$ and $a=1$.
Now let $t\mapsto f^t$ be a one-parameter semigroup in $\mathscr B[q;a]$, where $q\in \mathbb D$ and $a\in \mathbb{T}$. Consider the linear fractional transformation
$$
\begin{equation*}
L(z)=\frac{1-\overline{q}a}{a-q}\,\frac{z-q}{1-\overline{q}z}\,,
\end{equation*}
\notag
$$
which takes the unit disc $\mathbb D$ to itself and satisfies $L(q)=0$ and $L(a)=1$. For $t\geqslant 0$ set $\widetilde{f}^t(\zeta)=L\circ f^t \circ L^{-1}(\zeta)$. Clearly, $t\mapsto \widetilde{f}^t$ is a one-parameter semigroup of $\mathscr B[0;1]$. Hence, by the above its infinitesimal generator
$$
\begin{equation*}
\widetilde{v}(\zeta)= \frac{\partial}{\partial t}\widetilde{f}^t(\zeta)\bigg|_{t=0}
\end{equation*}
\notag
$$
admits a representation
$$
\begin{equation*}
\widetilde{v}(\zeta)=-\alpha\zeta (1-\zeta) \int_{\mathbb T} \frac{1-\varkappa}{1-\varkappa \zeta}\,d\mu(\varkappa),
\end{equation*}
\notag
$$
where $\alpha>0$ and $\mu$ is a probability measure on $\mathbb{T}$. On the other hand
$$
\begin{equation*}
\widetilde{v}\bigl(L(z)\bigr)= \frac{\partial}{\partial t}L\bigl(f^t(z)\bigr)\bigg|_{t=0}=L'(z)v(z),
\end{equation*}
\notag
$$
where $v$ is the infinitesimal generator of the one-parameter semigroup $t\mapsto f^t$. Therefore,
$$
\begin{equation*}
v(z)=-\alpha\frac{L(z)\bigl(1-L(z)\bigr)}{L'(z)}\int_{\mathbb T} \frac{1-\varkappa}{1-\varkappa L(z)}\,d\mu(\varkappa).
\end{equation*}
\notag
$$
Note that
$$
\begin{equation*}
\begin{gathered} \, \frac{L(z)\bigl(1-L(z)\bigr)}{L'(z)}= \frac{(z-q)(1-\overline{a}z)}{1-\overline{a}q}\,, \\ \frac{1-\varkappa}{1-\varkappa L(z)}= \frac{1-\overline{q}z}{1-\overline{a}q}\,\frac{1-\eta}{1-\eta\overline{a}z}\,, \end{gathered}
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\eta=\frac{(1-a\overline{q})\varkappa+\overline{q}(a-q)} {(1-\overline{a}q)+q(\overline{a}-\overline{q})\varkappa}
\end{equation*}
\notag
$$
ranges over the unit circle $\mathbb T$ as well as $\varkappa$ does. Then we obtain
$$
\begin{equation*}
v(z)=\frac{\alpha}{|1-\overline{a}q|^2}(q-z)(1-\overline{q}z)(1-\overline{a}z) \int_{\mathbb T}\frac{1-\eta}{1-\eta \overline{a}z}\,d\nu(\eta),
\end{equation*}
\notag
$$
where $\nu$ is a probability measure on $\mathbb{T}$. Thus, for an interior Denjoy–Wolff point the one-parameter semigroup $t\mapsto f^t$ in $\mathscr B[q;a]$ has an infinitesimal generator $v$ of the form specified in (7.8) and (7.2).
Turning to the case of a boundary Denjoy–Wolff point we assume first that $t\mapsto f^t$ is a non-trivial one-parameter semigroup in $\mathscr B[-1;1]$, so that the functions $f^t$, $t>0$, have the Denjoy–Wolff point $q=-1$ and fix the point $a=1$, at which they have finite angular derivatives $(f^t)'(1)$. By the Berkson–Porta formula (7.1) the infinitesimal generator $v$ of this one-parameter semigroup has a representation where $p_1$ is a holomorphic function in $\mathbb{D}$ with positive real part. However, this formula does not take account of the fact that the functions $f^t$ also have the fixed point $a=1$ in addition to $q=-1$.Just as in the case when $q=0$ and $a=1$, we have the equality $(f^t)'(1)=e^{\lambda t}$, $t\geqslant 0$, for some $\lambda >0$. Hence introducing the functions $g_t(z)=f^t\circ U_t(z)$, $t\geqslant 0$, we obtain
$$
\begin{equation*}
\frac{\partial}{\partial t}g_t(z)\bigg|_{t=0}=(1-z)^2p_2(z),
\end{equation*}
\notag
$$
where $p_2$ also is a holomorphic function in $\mathbb{D}$ with positive real part. Calculating this derivative otherwise we obtain In view of the form of the infinitesimal generator we can write this as follows:
$$
\begin{equation*}
(1-z)^2p_2(z)=\frac{\lambda}{2}(1-z^2)-(1+z)^2p_1(z).
\end{equation*}
\notag
$$
Setting here $z=0$ we obtain $p_1(0)+p_2(0)={\lambda}/{2}$. Thus we can apply the Riesz–Herglotz theorem to $p(z)=({2}/{\lambda})\bigl(p_1(z)+p_2(z)\bigr)$, which yields
$$
\begin{equation*}
p(z)=\int_{\mathbb T}\frac{1+\varkappa z}{1-\varkappa z}\,d\mu(\varkappa),
\end{equation*}
\notag
$$
where $\mu$ is a probability measure on $\mathbb{T}$. Now using the above inequalities we make several transformations:
$$
\begin{equation*}
\begin{aligned} \, v(z)&=(1-z)\biggl((1-z)p_2(z)-\frac{\lambda}{2}(1+z)\biggr) \\ &=(1-z)\biggl((1-z)\bigl(p_2(z)+p_1(z)\bigr)-\frac{\lambda}{2}(1+z)\biggr)- (1-z)^2p_1(z) \\ &=\frac{\lambda}{2}\,(1-z)\bigl((1-z)p(z)-(1+z)\bigr)+ \biggl(\frac{1-z}{1+z}\biggr)^2v(z). \end{aligned}
\end{equation*}
\notag
$$
Therefore,
$$
\begin{equation*}
\biggl(1-\biggl(\frac{1-z}{1+z}\biggr)^2\biggr)v(z)= -\lambda z(1-z)\int_{\mathbb T} \frac{1-\varkappa}{1-\varkappa z}\,d\mu(\varkappa),
\end{equation*}
\notag
$$
and after some obvious manipulations we arrive at the equality
$$
\begin{equation*}
v(z)=-\frac{\lambda}{4}(1+z)^2(1-z)\int_{\mathbb T} \frac{1-\varkappa}{1-\varkappa z}\,d\mu(\varkappa).
\end{equation*}
\notag
$$
Thus, for $q=-1$ and $a=1$ we also see that the infinitesimal generator has the form (7.8), (7.2).
It remains to consider the case when the one-parameter semigroup $t\mapsto f^t$ has the Denjoy–Wolff point $q$, and the fixed point $a$ is positioned arbitrarily on the unit circle $\mathbb{T}$. Using the linear fractional map
$$
\begin{equation*}
L(z)=\frac{q-(2-a\overline{q})z}{(q-2a)+a\overline{q}z}\,,
\end{equation*}
\notag
$$
which takes the unit disc $\mathbb{D}$ to itself and satisfies $L(q)=-1$ and $L(a)=1$, we define the family of functions $\widetilde{f}^t(\zeta)=L\circ f^t \circ L^{-1}(\zeta)$ for $t\geqslant 0$. It is easy to see that $t\mapsto \widetilde{f}^t$ is a one-parameter semigroup in $\mathscr B[-1; 1]$. By what we have proved, the infinitesimal generator of the one-parameter semigroup $t\mapsto \widetilde{f}^t$ has the form
$$
\begin{equation*}
\widetilde{v}(\zeta)= \frac{\partial}{\partial t}\widetilde{f}^t(\zeta)\bigg|_{t=0}= -\alpha(1+\zeta)^2(1-\zeta)\int_{\mathbb T} \frac{1-\varkappa}{1-\varkappa \zeta}\,d\mu(\varkappa),
\end{equation*}
\notag
$$
where $\alpha>0$ and $\mu$ is a probability measure on $\mathbb{T}$. However, the infinitesimal generator $v$ of the one-parameter group $t\mapsto f^t$ can be defined by the equality
$$
\begin{equation*}
v(z)=-\alpha\frac{\bigl(1+L(z)\bigr)^2\bigl(1-L(z)\bigr)}{L'(z)} \int_{\mathbb T}\frac{1-\varkappa}{1-\varkappa L(z)}\,d\mu(\varkappa).
\end{equation*}
\notag
$$
Note that
$$
\begin{equation*}
\frac{\bigl(1+L(z)\bigr)^2\bigl(1-L(z)\bigr)}{L'(z)}= \frac{4\alpha(q-z)(1-\overline{q}z)(1-\overline{a}z)} {(q-2a)+a\overline{q}z}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\frac{1-\varkappa}{1-\varkappa L(z)}= -\frac{(q-2a)+a\overline{q}z}{a|q-a|^2}\, \frac{1-\eta}{1-\eta\overline{a}z}\,,
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\eta=\frac{a\overline{q}+(2-a\overline{q})\varkappa} {\overline{a}q\varkappa+(2-\overline{a}q)}
\end{equation*}
\notag
$$
ranges over the circle $\mathbb{T}$ as well as $\varkappa$ does. Hence we have the equality
$$
\begin{equation*}
v(z)=\frac{4\alpha}{|q-a|^2}(q-z)(1-\overline{q}z)(1-\overline{a}z) \int_{\mathbb T}\frac{1-\eta}{1-\eta \overline{a}z}\,d\nu(\eta),
\end{equation*}
\notag
$$
where $\nu$ is a probability measure on $\mathbb{T}$.
As a result, we have shown that for any $q\in \overline{\mathbb{D}}$ and $a\in \mathbb{T}$ the infinitesimal generator $v$ of any one-parameter semigroup $t\mapsto f^t$ in $\mathscr B[q; a]$ has the form described by (7.8) and (7.2). Next we prove sufficiency. Let $v$ be a function of the form (7.8), (7.2), that is,
$$
\begin{equation*}
v(z)=\alpha(q-z)(1-\overline{q}z)(1-\overline{a}z)h(\overline{a}z),
\end{equation*}
\notag
$$
where $\alpha>0$, $q\in \overline{\mathbb{D}}$, $ a\in \mathbb{T}$, and $h$ is a function in the class $\mathscr Q$. By Corollary 7.1 we have
$$
\begin{equation*}
\operatorname{Re}\bigl((1-\overline{a}z)h(\overline{a}z)\bigr)\geqslant 0
\end{equation*}
\notag
$$
for $z \in \mathbb D$. This means that we can regard $v$ as the infinitesimal generator of a one-parameter semigroup $t\mapsto f^t$ in $\mathscr B[q]$. It remains to show that the functions $f^t$, $ t>0$, fix $a$ and have finite angular derivatives at this point.
First we show that $v$ itself has a finite angular derivative $v'(a)$ at $a$. For $M>1$ let
$$
\begin{equation*}
\Delta_M(a)=\biggl\{z\in \mathbb D\colon \frac{|a^2-z^2|}{1-|z|^2}<M\biggr\}
\end{equation*}
\notag
$$
be a lune with vertices $a$ and $-a$. From the standpoint of hyperbolic geometry in $\mathbb D$, which is usually defined by the length element the boundary $\partial\Delta_M(a)$ of the lune is the locus of points lying at a fixed distance from the diameter of $\mathbb D$ with endpoints $a$ and $-a$. Let
$$
\begin{equation*}
\Delta^+_M(a)=\bigl\{z\in \Delta_M(a)\colon \operatorname{Re}(\overline{a}z)>0\bigr\}
\end{equation*}
\notag
$$
denote the half of $\Delta_M(a)$ adjoining $a$. Using inequality (7.7) for functions in $\mathscr{Q}$ we obtain
$$
\begin{equation*}
\biggl|\frac{1-\varkappa}{1-\varkappa \overline{a}z}\biggr|\leqslant \frac{2|1-\overline{a}z|}{1-|z|^2}=\frac{2}{|1+\overline{a}z|}\, \frac{|a^2-z^2|}{1-|z|^2}\leqslant 2M
\end{equation*}
\notag
$$
for all $\varkappa \in \mathbb T$ and $z\in \Delta^+_M(a)$. Hence we can take the limit as $z \to a$, $z\in \Delta^+_M(a)$, in the integral defining $h$. This yields
$$
\begin{equation*}
\lim_{z \to a}h(\overline{a}z)=\lim_{z \to a}\int_{\mathbb T} \frac{1-\varkappa}{1-\varkappa \overline{a}z}\,d\mu(\varkappa)= \mu(\mathbb T\setminus\{1\}).
\end{equation*}
\notag
$$
Hence, as $z \to a$, $z\in \Delta^+_M(a)$, we have
$$
\begin{equation*}
\lim_{z \to a}v(z)=0\quad\text{and}\quad \lim_{z \to a}\frac{v(z)}{z-a}= \alpha|1-\overline{q}a|^2\mu(\mathbb T\setminus\{1\})=v'(a).
\end{equation*}
\notag
$$
Thus, the angular derivative $v'(a)$ exists and is finite. Moreover, $v'(a)>0$ only in the case when the measure $\mu$ is not concentrated at the point $\varkappa=1$. The latter case corresponds to $v(z)\equiv 0$ and $f^t(z)\equiv z$ for all $t\geqslant 0$. Thus we assume in what follows that $v'(a)>0$.
Let $R_a=\{z=ra\colon 1/2<r<1\}$ be the half-radius with endpoint $a$. Fix $M>1$ and choose $\delta>0$ less than the hyperbolic distance between $R_a$ and the boundary $\partial\Delta^+_M(a)$ of $\Delta^+_M(a)$. For each $z \in R_a$ let
$$
\begin{equation*}
\tau_z=\inf\{t>0\colon f^t(z)\in \partial\Delta^+_M(a)\}
\end{equation*}
\notag
$$
be the time of the first exit of $f^t(z)$ from $\Delta^+_M(a)$. Let $\gamma_z$ denote the curve $\zeta(t)=f^t(z)$, $0\leqslant t<\tau_z$. Using inequality (7.7), for $z\in \Delta^+_M(a)$ we obtain
$$
\begin{equation*}
\frac{|v(z)|}{1-|z|^2}\leqslant \frac{4\alpha\,|a-z|\,|h(\overline{a}z)|}{1-|z|^2}\leqslant \frac{8\alpha\,|a-z|\,|1-\overline{a}z|}{(1-|z|^2)^2}\leqslant \frac{ 8\alpha\,|a^2-z^2|^2}{(1-|z|^2)^2}\leqslant 8\alpha M^2.
\end{equation*}
\notag
$$
However, then the hyperbolic length $l_{\rm H}(\gamma_z)$ of $\gamma_z$ satisfies
$$
\begin{equation*}
l_{\rm H}(\gamma_z)=2\int_{\gamma_z}\frac{|d\zeta|}{1-|\zeta|^2}= 2\int_{0}^{\tau_z}\frac{\bigl|v\bigl(f^t(z)\bigr)\bigr|} {1-|f^t(z)|^2}\:dt \leqslant 16 \alpha M^2\tau_z.
\end{equation*}
\notag
$$
By the choice of $\delta$, for all $z \in R_a$ we have
$$
\begin{equation*}
\delta \leqslant l_{\rm H}(\gamma_z)\leqslant 16 \alpha M^2\tau_z,
\end{equation*}
\notag
$$
so that Thus, if $t \in [0,\tau^*]$ and $z \in R_a$, then the hyperbolic distance from $z$ to $f^t(z)$ is at most $16 \alpha M^2\tau^*=\delta$. Therefore, for $t \in [0,\tau^*]$ the image $f^t(R_a)$ of the radial segment $R_a$ lies in $\Delta^+_M(a)$, and $f^t(z)\to a$ as $z \to a$ along $R_a$.
Now since
$$
\begin{equation*}
\frac{\partial}{\partial t}\ln \frac{f^t(z)-a}{z-a}= \frac{v(f^t(z))}{f^t(z)-a}\,,
\end{equation*}
\notag
$$
where we consider the branch of the logarithm vanishing at $t=0$, it follows that
$$
\begin{equation*}
\ln\frac{f^t(z)-a}{z-a}=\int_0^t\frac{v(f^s(z))}{f^s(z)-a}\,ds.
\end{equation*}
\notag
$$
As shown above, the infinitesimal generator $v$ has a finite angular limit $v'(a)>0$ at $a$. Hence, taking the limit as $z \to a$ along $R_a$ in the above equality we obtain This means that the functions $f^t$, $0<t\leqslant \tau^*$, have angular derivatives at $a$ and In view of the semigroup property $f^{t+s}(z)=f^t\circ f^s(z)$ both the existence of the angular derivative at $a$ and the equality for it hold for all functions $f^t$, $t>0$, in the semigroup. Theorem 47 is proved. The next result yields a further detalization of the Berkson–Porta formula. Theorem 48 (Goryainov [75]). A holomorphic function $v$ in $\mathbb D$ is the infinitesimal generator of a one-parameter semigroup $t\mapsto f^t$ in $\mathscr B$ with Denjoy–Wolff point $q$, $q\in \overline{\mathbb D}$, and fixed points $a_1,\dots,a_n$ such that the functions $f^t$, $t>0$, have finite angular derivatives at these points if and only if it has a representation where $\lambda_k> 0$, $k=1,\dots,n$, and $p$ is a holomorphic function with non-negative real part in ${\mathbb D}$. 8. Conditions for embedding iterates in a one-parameter semigroupAs mentioned above, the problem of embedding the iterates of a function in a one-parameter semigroup is closely connected with the solutions to Abel’s and Schröder’s equations. In terms of solutions of these equations, conditions for embedding in $\mathscr B$ were obtained in [76]. The position of the Denjoy–Wolff point turns out to be important there. Now we present these results. Theorem 49 (Elin, Goryainov, Reich, and Shoikhet [76]). Let $f\in\mathscr B[0]$ be a function distinct from a linear fractional transformation of the disc $\mathbb{D}$, and assume that $f'(0)=\gamma \ne 0$. Then $f$ can be embedded in a one-parameter semigroup in $\mathscr B[0]$ if and only if the functional equation has a solution $F$ that is a holomorphic function in $\mathbb{D}$ such that where $p$ is a holomorphic function in $\mathbb{D}$ with positive real part and $e^{-p(0)}=\gamma$. Theorem 50 (Elin, Goryainov, Reich, and Shoikhet [76]). Let $f\in\mathscr B[1]$ be a function distinct from a linear fractional transformation of $\mathbb{D}$. Then $f$ can be embedded in a one-parameter semigroup in $\mathscr B[1]$ if and only if the functional equation has a solution $F$ that is a holomorphic function in $\mathbb{D}$ such that for $z\in\mathbb{D}$. Note that (8.1) is called Schröder’s equation and (8.2) is Abel’s equation. An extended literature is devoted to these functional equations. In the case when the Denjoy–Wolff point of a function $f$ in $\mathscr B$ is an interior point and $f'(q)=\gamma \ne 0$ there exists a limit which is a holomorphic function in $\mathbb{D}$ such that $K(q)=0$ and $K'(q)=1$. This function solves Schröder’s functional equation it is called the Koenigs function. The next result provides a full description of the Koenigs functions corresponding to functions $f$ embeddable in one-parameter semigroups in $\mathscr B[q]$, $q\in\mathbb{D}$.Theorem 51 (Goryainov and Kudryavtseva [74]). A holomorphic function $K$ in $\mathbb D$ is the Koenigs function for some function $f$ embeddable in a one-parameter semigroup in $\mathscr B[q]$, where $q\in \mathbb D$, if and only if it has a representation of the form for some $\sigma=e^{i\theta}$, $-\pi/2<\theta<\pi/2$, and some probability measure $\mu$ on the unit circle $\mathbb T$. Here the branches of the power function and logarithm that take the values $1$ and $0$, respectively, at $z=q$ are chosen. Note also that the function $K$ defined by (8.3) is univalent and takes $\mathbb{D}$ to a $\theta$-spiral domain. In turn, the family of functions $t\geqslant 0$, defines a one-parameter semigroup $t\mapsto f^t$ in $\mathscr B$ with Denjoy–Wolff point $q$. Thus, (8.3) defines the Koenigs functions of one-parameter semigroups in $\mathscr B$ with Denjoy–Wolff point $q$ in $\mathbb{D}$.In the case of a boundary Denjoy–Wolff point, for an embeddable function $f$ in the class $\mathscr B[q]$ we can also define its Koenigs function $F$, which solves Abel’s equation, is univalent in $\mathbb{D}$, and defines a one-parameter semigroup by the formula A full description of these function is provided by the following theorem. Theorem 52 (Goryainov and Kudryavtseva [74]). A holomorphic function $F$ in $\mathbb D$ is the Koenigs function of a one-parameter semigroup $t\mapsto f^t$ in $\mathscr B[q]$, where $q\in \mathbb{T}$, if and only if it has a representation A full description of the Koenigs functions in the case when, in addition to the Denjoy–Wolff point, the function also has other fixed points can be found in [75]. For a long time the interest in the problem of embedding the iterates of a holomorphic self-map of the unit disc in a on-parameter semigroup was prompted by the problem of embedding a Galton–Watson process in a homogeneous Markov branching process with continuous time. A Galton–Watson process is the simplest model of a branching process with discrete time. It describes the evolution of a population of identical particles that reproduce and die independently of one another, governed by some stochastic laws. Let $p_k$ be the probability of the event that a particle transforms into $k$ particles in the next generation, $\sum_{k=0}^{\infty}p_k=1$. The function $f(z)=\sum_{k=0}^{\infty}p_k z^k$ is called the generating function of the process; it determines the process fully. If at the initial moment of time the process starts with one particle, then $f$ can be regarded as the generating function of the size of the first generation of particles. The assumptions made in the definition of a Galton–Watson process imply that the generating function of the size of the $n$th generation is the $n$th iterate of $f$. Thus, the dynamics of a Galton–Watson process is described by the positive integer iterates of the probability generating function. Let $\mathscr B^+$ denote the set of probability generating functions. Note that $\mathscr B^+$ is a subsemigroup of the semigroup $\mathscr B$, and to each function $f$ in $\mathscr B^+$ we can uniquely assign a Galton–Watson process. The condition for embedding $f$ in a one-parameter semigroup in $\mathscr B^+$ is equivalent to the embeddability of the corresponding Galton–Watson process in a homogeneous Markov branching process with continuous time. Let $\widetilde{\mathscr B^+}$ denote the subset of probability generating functions that can be embedded in a one-parameter semigroup in $\mathscr B^+$. The first result on the description of the class $\widetilde{\mathscr B^+}$ were apparently due to Harris [77]. In [78] (Chap. V, § 5) he investigated this problem separately, in the context of fractional iterates of probability generating functions. In particular, he considered the case when $f(0)=0$, that is, $p_0=0$. He showed that if such $f$ is an entire function, then it cannot be embedded in a one-parameter semigroup of probability generating functions. Furthermore, he obtained conditions for embedding in the case when $f(0)=0$, which have the form of inequalities for certain recursive expressions involving Taylor coefficients of the functions $[f(z)]^n$, $n=1,2,\dots$ . Further steps in the solution of the embedding problem were made by Karlin and McGregor. In [79] they showed that if the probability generating function $f$ is meromorphic in $\mathbb{C}$, then it is not embeddable, unless it is linear fractional. Moreover, necessary conditions for embeddability were obtained in [79] in terms of certain relations for the first three derivatives of the probability generating function. A complete solution of the problem of embedding Galton–Watson processes in homogeneous Markov branching processes with continuous time was obtained in [80] and [81]. Now let $f$ be a probability generating function, that is, let $f \in \mathscr B^+$. Then $f$ takes the interval $[0,1]$ to itself, and its Denjoy–Wolff point $q=\lim_{n \to \infty}\!f^n(0)$ lies on this interval. In addition, $q$ is the probability that the corresponding Galton–Watson process degenerates. Also note that $z=1$ is a fixed point of any probability generating function $f$, and the angular derivative $f'(1)$ is the expected value of the number of descendants of a particle in the next generation of the corresponding Galton–Watson process. The following three theorems provide full descriptions, in different terms, of the class $\widetilde{\mathscr B^+}$ of embeddable probability generating functions. Theorem 53 (Goryainov [81]). Let $f \in \mathscr B^+$, and let $f(z) \not\equiv 1$. Then $f \in \widetilde{\mathscr B^+}$ if and only if the equation has a solution that is a holomorphic function in $\mathbb{D}$ such that $v'(0)=-1$ and $v^{(n)}(0) \geqslant 0$ for $n=2,3,\dots$ . Theorem 54 (Goryainov [80]). Let $f \in \mathscr B^+$, let $q$ be its Denjoy–Wolff point, and assume that $f'(1) \ne 1$. Then $f \in \widetilde{\mathscr B^+}$ if and only if $f'(z) \ne 0$ for $z \in \mathbb{D}$ and the locally uniform limit exists in $\mathbb{D}$ and satsifies $u^{(k)}(0) \geqslant 0$ for $k=2,3,\dots$ . Theorem 55 (Goryainov [80]). Let $f \in \mathscr B^+$ and $f'(1)=1$. Then $f \in \widetilde{\mathscr B^+}$ if and only if $f'(z) \ne 0$ for $z \in \mathbb{D}$ and the localy uniform limit exists in $\mathbb{D}$ and satisfies the conditions $u(f(0))=f'(0)$ and $u^{(k)}(0) \geqslant 0$ for $k=2,3,\dots$ . We also present a necessary condition for embeddability that can be stated in terms of the initial probabilities. Theorem 56 (Goryainov [81]). Let $f(z)=\sum_{k=0}^{\infty}p_kz^k \not\equiv z$ be a function in $\mathscr B^+$ with Denjoy–Wolff point $q$, $0 < q \leqslant 1$. Then the following hold: (a) if the point $(p_0, p_1)$ does not lie in the set
$$
\begin{equation*}
\biggl\{(x,y) \in \mathbb{R}^2 \colon \frac{1}{q}(q-x)(1-x) \leqslant y < \frac{1}{q}(q-x),\ 0 \leqslant x < q\biggr\},
\end{equation*}
\notag
$$
then $f \notin \widetilde{\mathscr B^+}$; (b) if $p_1=(q-p_0)(1-p_0)/q$ and $f \in \widetilde{\mathscr B^+}$, then $f$ is linear fractional. Integral representations for Koenigs functions corresponding to embeddable probability generating functions were obtained in [82]. 9. Angular derivative and Taylor coefficientsFor several decades the dominant topic of the theory of univalent functions was the coefficient problem; for univalent holomorphic functions $f(z)=z+c_2z^2+ \dotsb$ in the unit disc $\mathbb D$ it consists in determining the domain of values of the system of coefficients $\{c_2,\dots,c_n\}$ for each $n\geqslant 2$. A special case of this problem is to find sharp bounds for the coefficients. Bieberbach [39] conjectured that $|c_n|\leqslant n$. Attempts to prove or refute Bieberbach’s conjecture gave rise to the parametric method (see, for instance, [32]), which became a powerful tool in modern function theory. For instance, using this method Löwner [51] confirmed Bieberbach’s conjecture for the third coefficient, and Garabedian and Schiffer [83] proved it for the fourth coefficient. A complete solution to Bieberbach’s conjecture was found by de Branges [84], [85]; it was also based on Löwner’s parametric method. A great number of papers are devoted to describing the sets of values of coefficients or certain functionals of coefficients, on the class of univalent holomorphic functions in the unit disc and various subclasses of it. Estimates for the coefficients of univalent functions can be regarded as necessary conditions for univalence. The natural problem arising in this connection concerns finding necessary conditions for the existence of fractional iterates of a function that have the form of estimates for coefficients (because univalence is necessary for embedding the function in a one-parameter semigroup). Let $\mathscr B_r[0]$ be the set of functions $f$ in $\mathscr B[0]$ whose expansions in Taylor series have real coefficients:
$$
\begin{equation*}
\mathscr B_r[0]=\{f\in \mathscr B[0]\colon f^{(n)}(0) \in {\mathbb R},\ n=1,2,\dots\}.
\end{equation*}
\notag
$$
The statement below presents necessary conditions for embedding $f \in \mathscr B_r[0]$ in a one-parameter semigroup in $\mathscr B_r[0]$ which are expressed in terms of estimates for its first Taylor coefficients. Theorem 57 (Kudryavtseva [86]). Let $f(z)=c_1z+c_2z^2+\cdots$ be a function in $\mathscr B_r[0]$ that is embedded in a one-parameter semigroup in $\mathscr B_r[0]$. Then and The result of Theorem 57 can be put into a geometrically transparent form. To do this, we distinguish a set $\mathscr C(c_1)$ in $\mathbb R^2$ by
$$
\begin{equation*}
\mathscr C(c_1)=\{(c_2,c_3)\in \mathbb R^2 \colon f(z)=c_1z+c_2z^2+\cdots,\ f\in \mathscr E(\mathscr B_r[0])\},
\end{equation*}
\notag
$$
where $\mathscr E(\mathscr B_r[0])$ is the set of functions $f\in \mathscr B_r[0]$ describing the embedding in a one-parameter semigroup in $\mathscr B_r[0]$. Then Theorem 57 gives an analytic description of the boundary of $\mathscr C(c_1)$, which can be formulated in the following equivalent form. Theorem 57'. Fix $c_1\in (0,1)$. Then $\mathscr C(c_1)$ is a closed subset of $\mathbb R^2$ bounded by the curves and where $-2c_1(1-c_1)\leqslant c_2 \leqslant 2c_1(1-c_1)$. The question of whether the above bounds for the initial coefficients of $f\in \mathscr E(\mathscr B_r[0])$ are of interest can be solved by comparing them with the corresponding sharp estimates for the class of bounded univalent functions with real coefficients (see [87]). It turns out that $\mathscr C(c_1)$ lies in the corresponding set constructed starting from the sharp estimates for the class of bounded univalent functions with real coefficients. A description of extremal functions in the problem of bounds for the third coefficient in the class of functions $f\in \mathscr B_r[0]$ that admit en embedding in a one-parameter semigroup in $\mathscr B_r[0]$ is provided by the next theorem. Theorem 58 (Kudryavtseva [86]). Let $f(z)=c_1z+c_2z^2+\cdots$ be a function in $\mathscr B_r[0]$ that is distinct from the identity mapping and can be embedded in a one- parameter semigroup in $\mathscr B_r[0]$. Then the following assertions hold: (a) the equality holds if and only if $ f(z)=F^{-1}\bigl(c_1\,F(z)\bigr)$, where $F$ is the Koenigs function defined by
$$
\begin{equation*}
F(z)=\frac{z}{(1+z)^{1-\lambda}(1-z)^{1+\lambda}} \quad\textit{and}\quad \lambda=\frac{c_2}{2c_1(1-c_1)}\,;
\end{equation*}
\notag
$$
(b) the equality holds if and only if $f(z)=F^{-1}(c_1\,F(z))$, where $F$ is the Koenigs function defined byThe influence of the values of the angular derivatives at boundary fixed points on the domains of values of coefficients on the whole of $\mathscr B$ (making no assumption of univalence) is interesting to investigate. If we consider each coefficient separately, then generally speaking there can be nothing sharper than the bound $|c_n|\leqslant 1$. On the other hand, if we intend to find bounds for $c_n$ in the case when the first $n-1$ coefficients are fixed, then it follows from a result of Schur [88] that, within the class $\mathscr B[0]$, the range of $c_n$ is a disc with centre and radius depending on the previous coefficients $c_1,\dots,c_{n-1}$. Here are Schur’s inequalities for the first coefficients. Theorem 59 (Schur [88]). Let $f(z)=c_1z+c_2z^2+\dotsb$ be a function in the class $\mathscr B[0]$. Then Estimates in Theorem 59 are sharp; equalities are attained at certain Blaschke products. In [6] estimates for the joint ranges of coefficients in the class $\mathscr B[0,1]$ were obtained. We present the results for the first two coefficients. Theorem 60 (Goryainov [6]). Let $f(z)=c_1z+c_2z^2+\cdots$ be a function in the class $\mathscr B_{\alpha}[0,1]$, where $\alpha>1$. Then and Inequalities (9.1) and (9.2) are sharp, with equalities for certain Blaschke products. A method for deriving sharp inequalities for Taylor coefficients of functions with an arbitrary set of distinct fixed boundary points was developed in [89]. Let $\mathscr B[0,a_1,\dots,a_n]$ denote the set of functions in $\mathscr B[0]$ that fix distinct boundary points $z=a_j$, $|a_j|=1$, $j=1,\dots,n$, and have finite angular derivatives at these points:
$$
\begin{equation*}
\mathscr B[0,a_1,\dots, a_n]=\Bigl\{f\in \mathscr B[0]\colon \angle\lim_{z\to a_j}f(z)=a_j,\ \angle\lim_{z\to a_j}f'(z)= f'(a_j)<\infty\Bigr\}.
\end{equation*}
\notag
$$
The following theorem describes the set of values of the first two coefficients of functions in $\mathscr B[0,a_1,\dots,a_n]$. Theorem 61 (Kudryavtseva [89]). Let $f(z)=c_1z+c_2z^2+\cdots$ be a function in the class $\mathscr B[0,a_1,\dots,a_n]$, and let $f'(a_j)=\alpha_j$, $1\leqslant j\leqslant n$. Then Inequalities (9.3) and (9.4) are sharp, with equalities for certain Blaschke products. The first authors to deduce (9.3) were Cowan and Pommerenke [90]. However, one cannot fund bounds for higher coefficients using their method. Sharp estimates for all Taylor coefficients of the functions in $\mathscr B[0,a_1,\dots,a_n]$ rely on the following result of independent interest. Lemma 9.1 (Kudryavtseva). Let $f\in \mathscr B[0,a_1,\dots,a_k]$, $f(z)\not\equiv z$, and $f'(a_j)=\alpha_j$, $1\leqslant j\leqslant k$. Then the function belongs to the class $\mathscr B[0,a_1,\dots,a_{k-1}]$ and Proof. Let $g$ be the function defined by (9.5). Consider the composition $h\circ g(z)$, where $h(z)=(a_k+z)/(a_k-z)$ is the linear fractional map of the disc $\mathbb D$ onto the right-hand half-plane $\mathbb H=\{\zeta\in\mathbb C\colon \operatorname{Re}\zeta >0\}$. We can calculate its right-hand part:
$$
\begin{equation*}
\begin{aligned} \, \operatorname{Re} (h\circ g(z))&=\operatorname{Re}\frac{a_k+g(z)}{a_k-g(z)}= \frac{1}{\alpha_k-1}\operatorname{Re}\biggl(\alpha_k\frac{a_k+f(z)}{a_k-f(z)}- \frac{a_k+z}{a_k-z}\biggr) \\ &=\frac{1}{\alpha_k-1}\biggl(\alpha_k\frac{1-|f(z)|^2}{|a_k-f(z)|^2}- \frac{1-|z|^2}{|a_k-z|^2}\biggr). \end{aligned}
\end{equation*}
\notag
$$
Using the Julia–Carathéodory theorem to estimate the angular derivative $\alpha_k$, and using the inequality $\alpha_k>1$, which holds because $f$ is not the identity map and has an interior fixed point, we obtain $\operatorname{Re}(h\circ g(z))>0$ for all $z\in\mathbb D$. Then $g$ is a function in $\mathscr B$. Direct verification shows that $g$ fixes the points $z=0$ and $z=a_j$, $1\leqslant j\leqslant k-1$. In addition, we have formulae (9.6) for the angular derivatives of $g$ at boundary fixed points. $\Box$ In conclusion we look at the problem of the description of the domains of values of (individual) coefficients in the class $\mathscr B[0,1]$. Theorem 62 (Kudryavtseva and Solodov). Let $f(z)=c_1z+c_2z^2+\cdots$ be a function in the class $\mathscr B_{\alpha}[0,1]$, $\alpha>1$. Then and Inequalities (9.7) and (9.8) are sharp, with equalities attained at certain Blaschke products. For instance, we have equality in (9.8) for
$$
\begin{equation*}
f(z)=z\frac{(\alpha+1)z^2-(\alpha+1)z+2}{2z^2-(\alpha+1)z+\alpha+1}\,.
\end{equation*}
\notag
$$
Note that inequality (9.7) is equivalent to (9.1). We also observe that, as numerical experiments show, So we may conjecture that the set of values of the coefficient $c_n$ in the class $\mathscr B_{\alpha}[0,1]$, $\alpha>1$, is the horocycle at $1$ with parameter $\alpha+n-2$. Conjecture 1. Let $f(z)=c_1z+c_2z^2+\cdots$ be a function in $\mathscr B_{\alpha}[0,1]$, $\alpha>1$. Then As we mentioned already in § 6, the value $\alpha=2$ is critical: only for $\alpha<2$ do we have domains of univalence that contain both fixed points. For $\alpha=2$ the conjecture is quite concise: which is reminiscent of Bieberbach’s conjecture.
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\Bibitem{GorKudSol22} |
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