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This article is cited in 1 scientific paper (total in 1 paper) Sharp univalent covering domain for the class of holomorphic self-maps of a disc with fixed interior and boundary points O. S. Kudryavtsevaabc, A. P. Solodovab a Lomonosov Moscow State University, Moscow, Russia b Moscow Center of Fundamental and Applied Mathematics, Moscow, Russia c Volgograd State Technical University, Volgograd, Russia
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     § 1. IntroductionIn this paper we consider the properties of holomorphic self-maps of the unit disc $\mathbb D=\{z\in \mathbb C\colon |z|<1\}$ with fixed points. Such investigations have a rich history and open ways to the solution of rather complicated problems in the field of natural sciences where the dynamics of holomorphic maps is used in the description of some processes. One example is a problem in the theory of branching processes: embedding a Galton–Watson process in a homogeneous Markov branching process with continuous time. This is equivalent to the question of the existence of fractional iterates of the probability generating function of the discrete process, or, in other words, of embedding it in a one-parameter semigroup of probability generating functions. The infinite divisibility of a distribution law is equivalent to the embeddability of its characteristic function in a one-parameter multiplicative semigroup of characteristic functions. The description of such one-parameter semigroups is provided by the Lévy–Khintchine formula. In noncommutative probability theory, in connection with investigations of analogues of the Lévy–Khintchine formula, authors consider the problem of embedding the reciprocal Cauchy transform of a probability distribution in a one-parameter semigroup with respect to the operation of composition. (For a detailed discussion of problems in analysis and related areas where semigroups of holomorphic functions arise in a natural way, see [1].) Fixed points of a holomorphic self-map of the unit disc are important for the study of its properties. In the general case there need not exist fixed points in the interior of the unit disc. However, for each self-map we can distinguish the so-called Denjoy–Wolff point, which is the limit point of the sequence of (positive integer) iterations. If this is an interior point, then it is fixed. If this point lies on the boundary, then it is also fixed in the sense of angular limit. Apart from the Denjoy–Wolff point, there can also be other fixed points, but they must lie on the boundary of the unit disc. In his investigations of classes of holomorphic self-maps of the unit disc, Goryainov discovered [2] a new phenomenon: the existence of univalence domains under certain restrictions on the values of the angular derivatives at boundary fixed points. Subsequently, a search of sharp univalence domains was made for such classes in [3]–[7]. The next natural step is revealing the influence of angular derivatives on domains of univalent covering and describing their structure and size. These are the question we investigate in our paper. Generally speaking, the existence and the values of numerical characteristics of covering domains are a classical topic in geometric function theory. The result best known there is Koebe’s theorem on the existence of a single disc with centre at the origin that is covered by the values of an arbitrary univalent function $f$ in the unit disc $\mathbb D$ which is normalized by the conditions $f(0)=0$ and $f'(0)=1$. Koebe’s conjecture that this disc has radius $1/4$ was verified by Bieberbach. Theorem A (Koebe [8] and Bieberbach [9]). If $f$ is a univalent holomorphic function in the disc $\mathbb D$ satisfying the conditions $f(0)=0$ and $f'(0)=1$, then the image $f(\mathbb D)$ contains the disc $\{w\in\mathbb C\colon |w|<1/4\}$. In what follows we denote the class of holomorphic functions $f$ in the unit disc $\mathbb D$ that are normalized by the conditions $f(0)=0$ and $f'(0)=1$ by $\mathscr N$. Note that, of course, Koebe’s theorem fails on the whole of $\mathscr N$ (without the additional assumption that $f$ is univalent). For example, consider the functions $f(z)=(1-(1-z)^n)/n$, $n\in\mathbb N$: they do not take the value $1/n$ in $\mathbb D$, which can be arbitrarily close to zero for large $n$. At the same time, in place of univalence, we can impose other conditions on functions in $\mathscr N$. For example, the following result is due to Carathéodory. Theorem B (Carathéodory [10]). If a function $f\in \mathscr N$ has no zeros in the annulus $\{z\in\mathbb{D}\colon 0<|z|<1\}$, then $f(\mathbb D)$ contains the disc $\{w\in\mathbb C\colon |w|<1/16\}$. Nevertheless, when stated properly, the extremal problem of a covering disc has a nontrivial solution for the whole class $\mathscr N$. Bloch showed that for an arbitrary function in $\mathscr N$ there exists a disc of absolute radius (which, however, depends on the function) that is univalently covered by its values. Theorem C (Bloch [11]). There exists $\beta>0$ with the following properties. If ${f\in \mathscr N}$, then there exists a domain $\Delta\subset\mathbb D$ such that $f$ is univalent in $\Delta$ and the image $f(\Delta)$ contains a disc of radius $\beta$. The supremum $B$ of such $\beta$ is called the Bloch constant. The question of its value is an important unsolved problem in geometric function theory. The best upper bound known so far for the Bloch constant is due to Ahlfors and Grunsky [12]:
$$
\begin{equation*}
B\leqslant \frac{1}{\sqrt{1+\sqrt{3}}}\,\frac{\Gamma(1/3)\Gamma(11/12)}{\Gamma(1/4)}\approx{0,472}.
\end{equation*}
\notag
$$
Ahlfors [13] also found a lower bound which is in fact still the best one: $B\geqslant \sqrt{3}/4\approx{0,433}$. Subsequently, authors could only very slightly improve it. For example, Heins [14] showed that $B> \sqrt{3}/4$; after that Bonk [15] established the estimate $B> \sqrt{3}/4+10^{-14}$, while Chen and Gauthier [16] obtained the bound $B>\sqrt{3}/4+2\cdot 10^{-4}$. One of the first estimates for the Bloch constant is based on a result of independent interest concerning a disc of univalent covering for the class $\mathscr N_M$ of functions in $\mathscr N$ whose values are bounded by a constant $M>1$. Cauchy showed (see, for instance, [17]) that there exists a disc with centre at the origin that is covered univalently by all functions in $\mathscr N_M$, and he found the lower bound of $1/(6M)$ for its radius. Landau calculated the precise value of the radius of this disc. Theorem D (Landau [18]). Let $f\in\mathscr N_M$, $M>1$. Then $f$ covers univalently the disc $\{w\in\mathbb C\colon |w|<M(M-\sqrt{M^2-1})^2\}$. Furthermore, for each $R>{M(M-\sqrt{M^2-1})^2}$ there exists a function $f\in\mathscr N_M$ that does not cover univalently the disc $\{w\in\mathbb C\colon |w|<R\}$. It is interesting that if on a function in the class $\mathscr N$ we impose simultaneously the conditions of univalence and boundedness, then Theorems A and D can be improved as follows. Theorem E (Pick [19]). Let $f\in\mathscr N_M$, $M>1$, be a univalent function in $\mathbb D$. Then the image $f(\mathbb D)$ contains the disc $\{w\in\mathbb C\colon |w|<M(2M-1-2\sqrt{M(M-1)})\}$. It is easy to verify that, as $M\to 1$, the radius of the disc covered tends to $1$, while, as $M\to \infty$, this radius tends to $1/4$. Of course, the covering disc in Theorem E contains the one in Theorem D for each $M>1$. Note that under the assumptions of Theorem D we can abandon the assumption that the covering is univalent. In this case all functions in $\mathscr N_M$ cover a larger disc, whose size can be calculated precisely. Theorem F (Landau [20]). Let $f\in\mathscr N_M$, $M>1$. Then $f$ covers the disc $\{{w\in\mathbb C}$: $|w|<M \rho\}$, where $\rho$ is the root of the equation that lies between $0$ and $1$. Furthermore, for each $R>M \rho$ there exists a function $f\in\mathscr N_M$ that does not cover the disc $\{w\in\mathbb C\colon |w|<R\}$. The following result of Landau on the existence of a common univalence disc in the class $\mathscr N_M$ is particularly closely connected with Theorem D. Forestalling, we note that the same extremal functions establish the sharpness of Theorems D and G. Theorem G (Landau [18]). Let $f\in\mathscr N_M$, $M>1$. Then $f$ is univalent in the disc $\{z\in\mathbb D\colon |z|<M-\sqrt{M^2-1}\}$. Furthermore, for each $R>M-\sqrt{M^2-1}$ there exists a function $f\in\mathscr N_M$ that is not univalent in the disc $\{z\in\mathbb D\colon |z|<R\}$. Landau’s results on univalence and covering discs are often cited in geometric function theory and have numerous refinements and generalizations in various directions. For instance, note results due to Dieudonné [21] and Bermant [22] and related to $n$-valent functions. Theorem H (Dieudonné [21]). Let $f\in\mathscr N_M$, $M>1$. Then $f$ is at most $n$-valent in the disc $\{z\in\mathbb D\colon |z|<r_n\}$, where $r_n$ is the root of the equation that lies between $0$ and $1$. Furthermore, for each $R>r_n$ there exists a function $f\in\mathscr N_M$ that is at most $(n+1)$-valent in the disc $\{z\in\mathbb D\colon |z|<R\}$. Theorem I (Bermant [22]). Let $f\in\mathscr N_M$, $M>1$. Then $f$ covers the disc $\{{w\in\mathbb C}$: $|w|<M \rho_n\}$ with multiplicity at most $n$, where $\rho_n$ is the root of the equation that lies between $0$ and $1$. Furthermore, for each $R>M \rho_n$ there exists a function $f\in\mathscr N_M$ that does not cover the disc $\{w\in\mathbb C\colon |w|<R\}$ with multiplicity at most $n$. Theorem H covers Theorem G as its special case for $n=1$, and Theorem I links Theorems D and F, by filling the gap between a univalent covering and a covering with infinite multiplicity. Note that for any function in the class $\mathscr N_M$ a domain covered univalently by its values is much wider than the common disc of univalent covering. One will see from our results here that this does not only hold for individual functions, but also for rather wide subclasses of the class $\mathscr N_M$. Our aim is to find a sharp domain of univalent covering for one class important for applications, which consists of the holomorphic self-maps of the unit disc with fixed interior and boundary points that satisfy a certain restriction on the value of the angular derivative at the boundary fixed point. We will see below that, after a suitable normalization, the class of such functions lies in $\mathscr N_M$. The structure of the paper is as follows. In § 2 and § 3 we present revised proofs of Theorems G and D, which we believe to clarify the qualitative and quantitative aspects of these theorems and, apart from revealing the reasons for the existence of discs of univalence and univalent covering, also to explain the geometry underlying the particular sizes of these discs. In addition, we organize our arguments so that the search for the sharp disc of univalent covering becomes possible after finding the sharp univalence disc and is based on the explicit form of the latter. In § 4 we present the requisite information about fixed points of holomorphic maps and review briefly the classical and recent results on univalence domains in terms of the angular derivative at a boundary fixed point. A special place is given to the theorem on the sharp univalence domain for the class of holomorphic self-maps of the disc with fixed interior and boundary points with a bound on the value of the angular derivative at the boundary fixed point. This theorem plays a key role in our investigations. In § 5 we present the main result of the paper, a theorem on the sharp domain of univalent covering for the above class of functions. In its proof we use the approach demonstrated in § § 2 and 3; in accordance with this approach we use the explicit form of the univalence domain. In § 6 we find the disc of greatest radius that is univalently covered by all functions in the class under consideration (its centre is not at the origin, but is shifted towards the fixed point). In § 7 we extend the results from § 5 to the case of an arbitrary position of the interior fixed point. The results established in this paper were announced in [23] and [24]. § 2. New proof of Landau’s theorem on the univalence discIn this section we give a new proof of Landau’s theorem on the univalence disc for the class of bounded holomorphic functions in the unit disc $\mathbb D$. What are its differences from the original proof? In fact, Landau’s arguments (for instance, see [17]), apart from establishing the existence of the disc in which all functions in $\mathscr N_M$ are univalent, also allowed him to find the optimal value of its radius. Of course, this is a plus of the original proof, but this makes it more complicated. As concerns us, we assume that we know the radius of the common univalence disc. More precisely, we can guess it by calculating the univalence radius of the extremal function, namely, a certain Blaschke product of two factors. Using this assumptions, we avoid the technical difficulties of finding the precise value of the univalence radius and concentrate on the reasons for which a univalence disc must exist. We re-formulate Theorem G in terms of self-maps of the unit disc. But first we introduce the notation we need. Let $\mathscr B$ be the set of holomorphic functions mapping $\mathbb D$ to itself, and let $\mathscr B[0]$ denote the subclass of functions that fix $z=0$: For an arbitrary $M>1$, in $\mathscr B[0]$ we distinguish the subclass $\mathscr B_M[0]$ of functions whose derivative at $z=0$ has a modulus bounded away from zero by the quantity $1/M$:
$$
\begin{equation}
\mathscr B_{M}[0]=\biggl\{f\in \mathscr B[0]\colon |f'(0)|\geqslant\frac 1M\biggr\}.
\end{equation}
\tag{2.1}
$$
Taking the class $\mathscr B_{M}[0]$ in place of $\mathscr N_{M}$ we obtain the following equivalent statement of Theorem G. Theorem G$'$. Let $f\in \mathscr B_M[0]$, $M>1$. Then $f$ is univalent in the disc For any domain $\mathscr{V}$ such that $\mathscr L\subset\mathscr{V}\subset\mathbb D$, $\mathscr{V}\neq \mathscr L$, there exists a function $f\in \mathscr B_M[0]$ that is not univalent in $\mathscr{V}$. The proof of Theorem G$'$ — and therefore of Theorem G — is based on Landau’s and Löwner’s fundamental inequalities, on which we now dwell. By Schwarz’s lemma a function $f$ in $\mathscr B[0]$ maps each disc with centre at the origin to itself. Hence the fixed point $z=0$ is an attracting point for maps $f\in \mathscr B[0]$, except in the case of a rotation of $\mathbb D$. This attraction phenomenon turns out to be particularly pronounced if such a map takes two distinct points to the same point. Namely, the following inequality is due to Landau. Lemma A (Landau [18]). Let $f\in \mathscr B[0]$, and let $a, b\in \mathbb D$, $a\neq b$, be points such that $f(a)=f(b)=c$. Then Proof. Consider the composition of $f\in \mathscr B[0]$ with a linear-fractional self-map of $\mathbb D$: Clearly, $g\in \mathscr B$ and $g(a)=g(b)=0$. Then by the Schwarz–Pick lemma (see [25], Ch. VIII, § 1) $g$ can be represented in the following 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 a constant with modulus at most one. Setting $z=0$ in this representation we obtain $-c = a b h(0)$, which yields (2.3). The proof is complete. On the other hand points close to a fixed point cannot be attracted to it too strongly, as we see from the following result due to Löwner. Lemma B (Löwner [26]). Let $f\in \mathscr B[0]$ and $f(z)\not\equiv \varkappa z$ for $|\varkappa|=1$. Then for each $r\in(0,1)$ the image of the disc $\{z\in\mathbb D\colon |z|\leqslant r\}$ under the map $g(z)=f(z)/z$ lies in the non-Euclidean disc Proof. Since $f\in \mathscr B[0]$, by Schwarz’s lemma the function $g(z)=f(z)/z$ belongs to $\mathscr B$ and $g(0)=f'(0)$. Then by the Schwarz–Pick lemma, for each $z\in \mathbb D$, $z\neq 0$, we have the inequality
$$
\begin{equation*}
\biggl|\frac{g(z)-f'(0)}{1-\overline{f'(0)}g(z)}\biggr|\leqslant |z|.
\end{equation*}
\notag
$$
Hence if $|z|\leqslant r$, $r\in(0,1)$, then $g(z)$ lies in the non-Euclidean disc (2.4). The proof is complete. It is clear that if we choose sufficiently small $r$, then the non-Euclidean disc (2.4) does not contain the origin, which yields a lower bound for $|f(z)|/|z|$. Now we can proceed to the proof of the theorem. Proof of Theorem G$'$. First we estimate from above the univalence disc for the class $\mathscr B_M[0]$, establishing in this way the second part of the theorem. It is sufficient for the proof to present, for each boundary point of the disc $\mathscr L$ (see (2.2)), a function in $\mathscr B_M[0]$ whose derivative vanishes at this point. For each $\varkappa$, $|\varkappa|=1$, consider the Blaschke product It can easily be verified that for each $\varkappa$ the function $f_\varkappa$ belongs to $\mathscr B_M[0]$ and its derivative vanishes at $z_{\varkappa}=\varkappa(M-\sqrt{M^2-1})$.
Now we show that each function $f\in\mathscr B_M[0]$ is univalent in $\mathscr L$. If $f(z)=\varkappa z$, $|\varkappa|=1$, then its univalence is obvious. Assume that $f\in \mathscr B_M[0]$ is distinct from a rotation of $\mathbb D$ and is not univalent in $\mathscr L$, that is, there exist points $a, b\in \mathscr L$, $a\neq b$, such that $f(a)=f(b)$. Then by Lemma A we have $|f(a)|\leqslant |a|\, |b|<(M-\sqrt{M^2-1})|a|$, which is equivalent to the inequality where $ g(z)=f(z)/z$. In other words $g(a)$ must lie in $\mathscr L$.On the other hand, by Lemma B the function $g(z)$ takes $\mathscr L$ to the non-Euclidean disc
$$
\begin{equation}
\biggl\{w\in\mathbb D\colon \biggl|\frac{w-f'(0)}{1-\overline{f'(0)}w}\biggr|\leqslant M-\sqrt{M^2-1}\biggr\},
\end{equation}
\tag{2.6}
$$
which we now show to be disjoint from $\mathscr L$. In fact, (2.6) is the Euclidean disc $\{w\in\mathbb D\colon |w-c|=R\}$, where
$$
\begin{equation*}
c=\frac{\bigl(1-(M-\sqrt{M^2-1})^2\bigr)f'(0)}{1-(M-\sqrt{M^2-1})^2|f'(0)|^2}\quad\text{and} \quad R=\frac {(M-\sqrt{M^2-1})(1-|f'(0)|^2)}{1-(M-\sqrt{M^2-1})^2|f'(0)|^2}.
\end{equation*}
\notag
$$
For each $w$ in this disc and, in particular, for $g(a)$ we have the lower estimate
$$
\begin{equation}
|g(a)|\geqslant |c|-R =\frac{|f'(0)|-(M-\sqrt{M^2-1})}{1-(M-\sqrt{M^2-1})|f'(0)|}.
\end{equation}
\tag{2.7}
$$
Since $f\in\mathscr B_M[0]$, we have $|f'(0)|\geqslant 1/M$, and therefore
$$
\begin{equation}
\frac{|f'(0)|-(M-\sqrt{M^2-1})}{1-(M-\sqrt{M^2-1})|f'(0)|}\geqslant M-\sqrt{M^2-1}.
\end{equation}
\tag{2.8}
$$
From (2.7) and (2.8) we obtain Estimates (2.5) and (2.9) are in contradiction, so our assumption that $f\in\mathscr B_M[0]$ is not univalent in $\mathscr L$ fails. The proof is complete. Remark 1. From the proof of the theorem one understands very clearly the geometric interpretation of the precise value of the univalence radius as the instant at which the discs from Lemmas A and B touch one the other. In fact, the discs $\{w\in\mathbb D\colon|w|<r\}$ and $\bigl\{w\in\mathbb D\colon \bigl|(w-f'(0))/(1-\overline{f'(0)}w)\bigr|\leqslant r\bigr\}$ are disjoint for small $r$. As $r$ grows, the distance between them reduces, and they become tangent precisely as $r$ attains the value equal to the univalence radius (see (2.5), (2.6) and Figure 1). § 3. New proof of Landau’s theorem on the univalent covering discAs already mentioned in the introduction, it was Cauchy who discovered that there exists a disc covered univalently by all functions in $\mathscr{N}_M$. He also found a lower estimate for its radius: $R(M)\geqslant 1/(6M)$. Subsequently, Landau refined Cauchy’s techniques and found the precise value of the univalent covering radius: $R(M)=M(M-\sqrt{M^2-1})^2$ (for instance, see [17]). Landau established this result (Theorem D) independently of the result on the univalence disc (Theorem G). Moreover, first he found the precise radius of the univalent covering disc and only then the precise radius of the univalence disc. The new proof of the theorem of the univalent covering disc that we present here is formally also independent of the result on the univalence disc, but in fact it is based on it. As in the proof of Theorem G$'$, we do not calculate the precise value of the univalent covering radius, which we assume to be known. As the two theorems are intimately related, it is easy to believe that in both cases the same Blaschke products are extremal functions, so that we can also guess the precise value of the univalent covering radius: $R(M)=M(M-\sqrt{M^2-1})^2$. Finally, the core of the proof of Theorem D is the sharp univalence disc found in Theorem G. Just this disc contains the domain that a function in $\mathscr N_M$ maps on the disc of radius $R(M)$. As with Landau’s theorem on the univalence disc, it is convenient to state and prove Landau’s theorem on the univalent covering disc in terms of the class $\mathscr B_M[0]$. Theorem E$'$. Let $f\in \mathscr B_M[0]$, $M>1$. Then there exists a function inverse to $f$ and mapping the disc conformally onto a domain $\mathscr X\subset \mathbb D$. For any domain $\mathscr V$, $\mathscr W\subset \mathscr V \subset \mathbb D$, $\mathscr V\neq \mathscr W$, there exists a function $f\in \mathscr B_M[0]$ that has no inverse function in $\mathscr V$. Proof. As in the proof of Theorem G$'$, we begin with an upper estimate for the univalent covering disc, thus establishing the second part of the theorem. For each $\varkappa$, $|\varkappa|=1$, consider the Blaschke product For each $\varkappa$ the function $f_\varkappa$ belongs to $\mathscr B_M[0]$ and has derivative zero at $z_{\varkappa}=\varkappa (M-\sqrt{M^2-1})$. Hence the inverse function of $f_\varkappa$ has the branch point $w_{\varkappa}=f_{\varkappa}(z_{\varkappa})=\varkappa (M-\sqrt{M^2-1})^2$.
Now we show that all functions in $\mathscr B_M[0]$ are invertible in $\mathscr W$. Let $f\in \mathscr B_M[0]$. It is sufficient to verify that the equation $f(z)=w$ has a unique solution in some subdomain of $\mathbb D$ for each $w\in \mathscr W$. We show that we can take the disc $\mathscr L$ (see (2.2)) as this subdomain. For $z$ on the circle $\{z\in\mathbb D\colon |z|=M-\sqrt{M^2-1}\}$, by Lemma B the values of the function $g(z)=f(z)/z$ lie in the non-Euclidean disc
$$
\begin{equation*}
\biggl\{\zeta\in\mathbb D\colon \biggl|\frac{\zeta-f'(0)}{1-\overline{f'(0)}\zeta}\biggr|\leqslant M-\sqrt{M^2-1}\biggr\}.
\end{equation*}
\notag
$$
In the same way as in the proof of Theorem G$'$ we deduce the lower estimate
$$
\begin{equation*}
|g(z)|\geqslant \frac{|f'(0)|-(M-\sqrt{M^2-1})}{1-(M-\sqrt{M^2-1})|f'(0)|}.
\end{equation*}
\notag
$$
Bearing in mind that $|f'(0)|\geqslant 1/M$ we obtain $|g(z)|\geqslant M-\sqrt{M^2-1}$. Finally, we recall that $g(z)=f(z)/z$ and $|z|=M-\sqrt{M^2-1}$, and for $z$ in question we obtain the estimate $|f(z)|\geqslant (M-\sqrt{M^2-1})^2$. In other words, once $z$ occurs on the boundary of $\mathscr L$, the value $f(z)$ must occur in the interior of the disc
$$
\begin{equation*}
\mathscr K(z)=\biggl\{\zeta\in\mathbb D\colon \biggl|\frac{\zeta-f'(0)z}{z-\overline{f'(0)}\zeta}\biggr|\leqslant M-\sqrt{M^2-1} \biggr\},
\end{equation*}
\notag
$$
which is disjoint from $\mathscr W$.
In Figure 2 we show the mutual position of the discs in the case when ${|f'(0)|=1/M}$ (so that $\mathscr K(z)$ touches $\mathscr W$). As $z$ travels along the circle $\{z\in\mathbb D\colon |z|=M-\sqrt{M^2-1}\}$, the disc $\mathscr K(z)$ rolls around $\mathscr W$. Thus, as $z$ makes an anticlockwise rotation along the circle $\{z\in\mathbb D\colon |z|=M- \sqrt{M^2-1}\}$, its image $f(z)$ winds around $\mathscr W$, so that 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 $\mathscr L$. The proof is complete. Remark 2. The curve with the property described above (the discs $\mathscr K(z)$ corresponding to its points are disjoint from $\mathscr W$) is unique: it can only be the boundary of the disc $\mathscr L$. If $|f'(0)|=1/M$ and $|z|\neq M-\sqrt{M^2-1}$, then the discs $\mathscr K(z)$ and $\mathscr W$ intersect. Remark 3. Geometrically, the boundary of $\mathscr W$ can be characterized as the envelope of the family of the discs $\mathscr K(z)$ corresponding to points $z$ on the boundary of $\mathscr L$. Remark 4. We see from the proof that the statement of the theorem can be refined as follows: the domain $\mathscr X$ onto which a branch of the inverse function of $f\in \mathscr B_M[0]$ maps the disc $ \mathscr W$ conformally can be taken inside $\mathscr L$. § 4. Fixed points and univalence domainsThe study of the properties of a holomorphic self-map of a disc is closely connected with examining its fixed points. If $f\in \mathscr B$ and $f(z)\not\equiv z$, then by the Schwarz–Pick lemma the function $f$ can have at most one fixed point in ${\mathbb D}$. In general, $f\in \mathscr B$ does not necessarily have fixed points in ${\mathbb D}$. However, the Denjoy–Wolff theorem (see [17], Ch. VI, § 43) states that if $f \in \mathscr B$ is not a linear fractional automorphism of ${\mathbb D}$, then there exists a unique point $q$, $|q|\leqslant 1$, such that the sequence of iterates $f^{n}=f\circ f^{n-1}$, $n=2,3,\dots$, of $f=f^1$ converges to $q$ locally uniformly in ${\mathbb D}$. Moreover, if $q$ is a boundary point, that is, it lies on the unit circle $\mathbb{T} = \bigl\{ z\in\mathbb{C}\colon |z| = 1\bigr\}$, then the angular limits
$$
\begin{equation*}
\angle\lim_{z\to q}f(z)=f(q)\quad\text{and} \quad \angle \lim_{z\to q}f'(z)=\angle\lim_{z\to q}\frac{f(z)-q}{z-q}=f'(q)
\end{equation*}
\notag
$$
exist at this point, and we have $f(q)=q$ and $0<f'(q)\leqslant 1$. The point $q$ is called the Denjoy–Wolff point of $f$. If $q\in \mathbb D$, then by the Schwarz–Pick lemma we have $|f'(q)|\leqslant 1$. Thus the Denjoy–Wolff point of $f$ is an attracting ($|f'(q)|< 1$) or a neutral ($|f'(q)|= 1$) fixed point. If $f\in \mathscr B$ with Denjoy–Wolff point $q$, $|q|\leqslant 1$, has an additional fixed point $a$, then $a$ lies on the unit circle $\mathbb T$ and is fixed in the sense of the angular limit The angular limit always exists at the boundary fixed point (see, for instance, [27]). If it is finite, then it is positive and $f'(z)$ has the same angular limit as $z\to a$. In this case we call the limit in (4.1) the angular derivative of $f$ at $a$ and denote it by $f'(a)$.Let $\mathscr B\{1\}$ denote the class of functions with fixed point $a=1$:
$$
\begin{equation*}
\mathscr B\{1\}=\Bigl\{f\in \mathscr B\colon \angle \lim_{z\to 1} f(z)=f(1)=1\Bigr\}.
\end{equation*}
\notag
$$
Then the following classical inequality holds for the angular derivative. The following condition for local univalence at an interior point is well known: a function whose derivative at a point is distinct from zero is univalent in some neighbourhood of this point. As Valiron showed, for a boundary fixed point local univalence holds under a similar condition: a function with finite angular derivative at a boundary fixed point is univalent in a sector with opening arbitrarily close to $\pi$. Theorem K (Valiron [17]). 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 Baker and Pommerenke [28], [29] investigated in detail the shapes and sizes of univalence domains of a function $f\in\mathscr B\{1\}$ in their dependence on the behaviour of the iterates of this function. It turns out, in particular, that if the angular derivative is equal to $1$, then univalence can be ensured not just in a sector, but even in a domain with boundary tangent to the circle $\mathbb T$ at $z=1$. Fairly recently Becker and Pommerenke, in their investigations of univalence domains of functions in $ \mathscr B\{1\}$, obtained the following result. Theorem L (Becker and Pommerenke [30]). Let $f\in \mathscr B\{1\}$ and $f'(1)<\infty$. Then $f$ is univalent in the domain By the Julia–Carathéodory theorem the left-hand side of the inequality in (4.3) is always greater than $1$. Moreover, it tends to $1$ inside each angle, so that it is less than 2 inside the part of a sufficiently small neighbourhood that lies in this angle. Thus, Theorem L characterizes quantitatively Valiron’s result on univalence in a neighbourhood of a boundary fixed point. As the sector (4.2) inside which $f$ is univalent we can take a sector lying in the domain (4.3). By analogy with an interior fixed point, for any $\alpha>1$, in the class $\mathscr B\{1\}$ we select the subclass $\mathscr B_{\alpha}\{1\}$ of functions satisfying a restriction on the value of the angular derivative at the boundary fixed point:
$$
\begin{equation*}
\mathscr B_{\alpha}\{1\}=\bigl\{f\in \mathscr B\{1\}\colon f'(1)\leqslant \alpha\bigr\}.
\end{equation*}
\notag
$$
It was shown in [3] that a single condition on the angular derivative is not sufficient for the existence of a nonempty univalence domain. Theorem M (Kudryavtseva and Solodov [3]). For no $\alpha>1$ does there exist a nonempty univalence domain for the class $\mathscr B_{\alpha}\{1\}$. In other words, there is no full analogue of Landau’s theorem (Theorem G) for a boundary fixed point. The class $\mathscr B_{\alpha}\{1\}$ is too wide, and we must look at more narrow classes for nonempty univalence domains. The most natural reduction of the class $\mathscr B_\alpha\{1\}$ is its subclass of functions which, apart from the repelling fixed point $z=1$, have an attracting fixed point $z=q$. By the Denjoy–Wolff theorem an attracting fixed point exists in any case, and our additional condition consists in specifying its position. Consider the case when the Denjoy–Wolff point is interior. We assume without loss of generality that it is at the origin. We denote the class obtained by
$$
\begin{equation*}
\mathscr B_\alpha[0,1]=\mathscr B_\alpha\{1\}\cap \mathscr B[0], \qquad \alpha>1.
\end{equation*}
\notag
$$
This class of functions was thoroughly studied by Goryainov [2]. He showed, in particular, that for $\alpha\in(1,2)$ the class $\mathscr B_{\alpha}[0,1]$ lies in $\mathscr B_{\alpha/(2-\alpha)}[0]$ (see (2.1)). In view of this embedding Theorem G$'$ implies that there exists a common univalence domain for the class $\mathscr B_{\alpha}[0, 1]$, $\alpha\in(1,2)$. As such a domain we can take, for example, the disc with centre zero such that all functions in $\mathscr B_{\alpha/(2-\alpha)}[0]$, $\alpha\in(1,2)$, are univalent in it. Of course, this disc is not an optimal univalence domain; furthermore, as the conditions are not symmetric, the optimal domain has a very different geometry. In [2] a univalence domain was found which is larger than the disc obtained immediately from embeddings of classes of functions and Theorem G$'$. This domain does not only contain the interior fixed point $z=0$, but it also adjoins the boundary fixed point $z=1$. The further investigations (see [3], [6] and [7]) were related to the search of a maximal possible domain of univalence for the class $\mathscr B_{\alpha}[0, 1]$ for $\alpha\in(1,2)$ and to the question of the existence of a univalence domain for $\alpha\geqslant 2$, when there is no embedding relation between classes of functions and we cannot use Theorem G$'$. The difficulty was in the fact that, in contrast to the analogous problem for $\mathscr N_M$, not just the size of the possible maximal univalence domains was unknown, but rather its geometric shape. In [6] an [7] a sharp univalence domain was found for the class $\mathscr B_{\alpha}[0, 1]$ for $\alpha\in(1,4]$ (for $\alpha\in[2,4] $ the univalence domain does not contain the interior fixed point). The question of the existence and size of univalence domains for $\alpha>4$ still remains open. Theorem N (Solodov; see [6] and [7]). Let $f\in \mathscr B_{\alpha}[0, 1]$, $\alpha\in (1,4]$. Then $f$ is univalent in the domain Whatever the domain $\mathscr{V}$ such that $\mathscr D(\alpha)\subset\mathscr{V}\subset\mathbb D$, $\mathscr V\neq \mathscr D(\alpha)$, there exists a function $f\in \mathscr B_{\alpha}[0, 1]$ that is not univalent in $\mathscr{V}$. In Figure 3 we can see the shape of $\mathscr D(\alpha)$ and compare its size with the disc $\mathscr O(\alpha)$, in which all functions $f\in \mathscr B_{\alpha}[0, 1]$, $\alpha\in(1,2)$, are univalent by Theorem G$'$. The proof of Theorem N is technically rather complicated; it is based on the ideas discussed in § 2. The argument rests on the following inequality, which is in a certain sense similar to Landau’s (see Lemma A). Lemma C (Solodov; see [6] and [7]). Let $f\in \mathscr B\{1\}$, $f(0)=0$, and let $a, b\in \mathbb D$, $a\neq b$, be points such that $f(a)=f(b)=c$. Then where $\lambda(z)=-z{(1-\overline{z})}/{(1-z)}$. Remark 6. Since $|\lambda (z)|=|z|$ for each $z\in\mathbb D$, by Lemma A we have the estimate $|\lambda (c)/(\lambda(a)\lambda(b))|\leqslant 1$. Hence the last term on the right-hand side of (4.5) is nonnegative. Without going into technical details of the proof of Theorem N, note that for fixed $f'(1)\in (1,4]$ some domains in $\mathbb D$ have the following property: if $a$ and $b$ are distinct points in this domain, then (4.5) does not hold for any $c\in\mathbb D$. The existence of such domains follows from the properties of the involution $\lambda(z)$ (see [7]). The widest domain with this property is $\mathscr D(\alpha)$ (see (4.4)). The fact that $\mathscr D(\alpha)$ is sharp is ensured by the family of Blaschke products. For each $\theta\in(-\pi, \pi)$ consider the function
$$
\begin{equation*}
f_{\theta}(z)=z\frac{1-(\alpha-1)e^{i\theta}+\alpha e^{i\theta}z}{\alpha-(\alpha-1-e^{i\theta})z}.
\end{equation*}
\notag
$$
It is easy to verify that $f_{\theta}$ belongs to the class $\mathscr B_{\alpha}[0, 1]$ and its derivative vanishes at the point
$$
\begin{equation}
z_{\alpha}(\theta)= \frac{\sqrt{\alpha-1}-e^{-i\theta/ 2}}{\sqrt{\alpha-1}+e^{i \theta/ 2}}.
\end{equation}
\tag{4.6}
$$
It is also easy to see that (4.6) defines the curve bounding the domain $\mathscr D(\alpha)$.
§ 5. Sharp univalent covering domainThe existence of a nonempty univalent covering domain for the class $\mathscr B_{\alpha}[0,1]$, $\alpha\in(1,2)$, follows from the embedding of this class in $\mathscr B_{\alpha/(2-\alpha)}[0]$ and Theorem E$'$. Proposition 1. Let $f\in \mathscr B_{\alpha}[0, 1]$, $\alpha\in (1,2)$. Then there exists a function inverse to $f$ and mapping the disc conformally onto a domain $\mathscr X\subset \mathbb D$. For each $\alpha\in(1,2)$, $\mathscr{E}(\alpha)$ has the greatest radius among all discs with centre zero which are univalently covered by the values of each function in $\mathscr B_{\alpha}[0, 1]$. In fact, the function is in the class $\mathscr B_{\alpha}[0, 1]$ and its derivative vanishes at the point $z_{\alpha}=-(1-\sqrt{\alpha-1})/(1+\sqrt{\alpha-1})$. Hence the inverse function of $f$ has the branch point $w_\alpha=f(z_{\alpha})=-(1-\sqrt{\alpha-1})^2/(1+\sqrt{\alpha-1})^2$, which lies on the boundary of $\mathscr{E}(\alpha)$.Since the class $ \mathscr B_{\alpha}[0, 1]$ is much more narrow than $\mathscr B_{\alpha/(2-\alpha)}[0]$, we can expect for it larger univalence covering domains than the discs $\mathscr{E}(\alpha)$, especially since the above example only hinders extending a domain of univalent covering in the direction of one ray. The following theorem answers the question of the sharp universal covering domain for the class of functions with an interior and a boundary fixed point and with restrictions on the value of the derivative at the boundary fixed point. Theorem 1. Let $f\in \mathscr B_{\alpha}[0, 1]$, $\alpha\in(1,2)$. Then there exists a function inverse to $f$ that maps the domain conformally onto a domain $\mathscr X\subset \mathbb D$. Whatever the domain $\mathscr V$ such that $\mathscr W(\alpha)\subset \mathscr V \subset \mathbb D$, $\mathscr V\neq \mathscr W(\alpha)$, there exists a function $f\in \mathscr B_{\alpha}[0, 1]$ without an inverse function in $\mathscr V$. Finally, for $\alpha\geqslant 2$ there exist no nonempty domains of univalent covering for the class $\mathscr B_{\alpha}[0, 1]$. Remark 7. The domain $\mathscr W(\alpha)$ is bounded by the piecewise smooth curve with the equation Proof of Theorem 1. First we give an upper estimate for the domain of univalent covering, thus proving the second part of the theorem. For each $\theta\in (-\pi,\pi)$ consider the Blaschke product
$$
\begin{equation*}
f_{\theta}(z)=z\frac{1-(\alpha-1)e^{i\theta}+\alpha e^{i\theta}z}{\alpha-(\alpha-1-e^{i\theta})z}.
\end{equation*}
\notag
$$
The function $f_\theta$ belongs to the class $\mathscr B_{\alpha}[0, 1]$ and has a derivative vanishing at the point $z_{\theta}=({\sqrt{\alpha-1}-e^{-i\theta/ 2}})/({\sqrt{\alpha-1}+e^{i \theta/ 2}})$. Hence the inverse function of $f_\theta$ has the branch point $w_{\theta}=f_{\theta}(z_{\theta})=-e^{i\theta}({\sqrt{\alpha-1}-e^{-i\theta/ 2}})^2/({\sqrt{\alpha-1}+e^{i \theta/ 2}})^2$. Thus, each boundary point of $\mathscr W(\alpha)$ (see (5.3)) except $z=1$ is a branch point of a function inverse to some function in $\mathscr B_{\alpha}[0, 1]$.
Now we show that $\mathscr W(\alpha)$ is univalently covered by all functions in the class $\mathscr B_{\alpha}[0, 1]$. We select some function $f\in \mathscr B_{\alpha}[0, 1]$. It is sufficient 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 claim that we can take $\mathscr D(\alpha)$ as such a subdomain (see (4.4)). We implement the same plan of the proof as in Landau’s theorem on the universal covering disc (see Theorem E$'$) . The idea is to complete a circuit along the boundary of $\mathscr D(\alpha)$ and use the argument principle. By contrast to the boundary of the disc $\mathscr L$ (see (2.2)), the boundary of $\mathscr D(\alpha)$ contains a corner point $z=1$, at which $f$ is in general not necessarily analytic. However, this hindrance is easy to overcome: to do this, in the proof we replace a small piece of the boundary of $\mathscr D(\alpha)$ in a neighbourhood of the corner point by a suitable smooth curve lying fully in the disc $\mathbb D$. We do this in what follows, but first we verify that if a point $z$ lies on the curve
$$
\begin{equation}
z_\varphi=\frac{\sqrt{\alpha-1}-e^{-i \varphi/2}}{\sqrt{\alpha-1}+e^{i \varphi/2}}, \qquad \varphi\in(-\pi, \pi),
\end{equation}
\tag{5.4}
$$
then the value $f(z)$ cannot occur in the interior of $\mathscr W(\alpha)$. In fact, (5.4) consists precisely of the boundary points of $\mathscr D(\alpha)$, with the exception of one point occurring on $\mathbb T$, at which we do not assume $f$ to be analytic.
By Schwarz’s lemma $g(z)=f(z)/z$ is a function in $\mathscr B$. Moreover, it is easy to verify that $g\in \mathscr B_{\alpha-1}\{1\}$. By the Julia–Carathéodory theorem the value $g(z)$ satisfies the inequality
$$
\begin{equation*}
\frac{|1-g(z)|^2}{1-|g(z)|^2}\leqslant (\alpha-1)\frac{|1-z|^2}{1-|z|^2}.
\end{equation*}
\notag
$$
Thus $f(z)$ must lie in the disc
$$
\begin{equation*}
\mathscr U(z)=\biggl\{\zeta\in\mathbb D\colon 0\leqslant\frac{|z-\zeta|^2}{|z|^2-|\zeta|^2}\leqslant (\alpha-1)\frac{|1-z|^2}{1-|z|^2} \biggr\}
\end{equation*}
\notag
$$
for all $z\in\mathbb D$, including in the case when $z$ occurs on the curve (5.4). (For $\zeta=z$ we set $|z-\zeta|^2/\bigl(|z|^2-|\zeta|^2\bigr)=0$.)
Supposing that $f(z)\in\mathscr W(\alpha)$ for some $z$ on the curve (5.4), there exists at least one boundary point of $\mathscr W(\alpha)$ that lies in the interior of $\mathscr U(z)$. In other words, for some $\varphi,\theta\in(-\pi,\pi)$ a pair of points $z$, $\zeta$ of the form (5.4) and (5.3), respectively, satisfies the inequality
$$
\begin{equation}
\begin{aligned} \, \notag 0&\,{<}\,\frac{|(\!\sqrt{\alpha\,{-}\,1}{\kern1pt}{-}\,e^{-i \varphi/2})/(\!\sqrt{\alpha\,{-}\,1}{\kern1pt}{+}\,e^{i \varphi/2})\,{+}\,e^{i\theta} ((\!\sqrt{\alpha\,{-}\,1}{\kern1pt}{-}\,e^{-i\theta/2})/ (\!\sqrt{\alpha\,{-}\,1}{\kern1pt}{+}\,e^{i\theta/2}))^2|^2} {|(\sqrt{\alpha\,{-}\,1}\,{-}\,e^{-i \varphi/2})/(\sqrt{\alpha\,{-}\,1}\,{+}\,e^{i \varphi/2})|^2 \,{-}\,|(\sqrt{\alpha\,{-}\,1}\,{-}\,e^{-i\theta/2})/(\sqrt{\alpha\,{-}\,1}\,{+}\,e^{i\theta/2})|^4} \\ &\,{<}\,(\alpha-1)\frac{|1-(\sqrt{\alpha-1}-e^{-i \varphi/2})/(\sqrt{\alpha-1}+e^{i \varphi/2})|^2}{1-|(\sqrt{\alpha-1}-e^{-i \varphi/2})/(\sqrt{\alpha-1}+e^{i \varphi/2})|^2}. \end{aligned}
\end{equation}
\tag{5.5}
$$
After a sequence of transformations, (5.5) can be expressed in an equivalent form as follows:
$$
\begin{equation}
\begin{aligned} \, \notag 0&<\frac{(\alpha-1)(\alpha\cos(\theta/2)-2\cos(\varphi/2))^2 +((\alpha-1)\sin((\varphi+\theta)/2) +\sin((\varphi-\theta)/2))^2}{\sqrt{\alpha-1} (2\alpha^2\cos(\theta/2)-(\alpha^2+2\alpha-2)\cos(\varphi/2) -2(\alpha-1)\cos(\varphi/2)\cos\theta)} \\ &<\sqrt{\alpha-1}\cos\frac\varphi2. \end{aligned}
\end{equation}
\tag{5.6}
$$
Finally, equivalent transformations of (5.6) take us to the inequality
$$
\begin{equation}
8\alpha\sin^2\frac{\varphi-\theta}{4}\biggl( (\alpha-1)(\alpha+2)\sin^2\frac{\varphi+\theta}{4}+(2-\alpha) \cos^2\frac{\varphi-\theta}{4} \biggr)<0.
\end{equation}
\tag{5.7}
$$
Since $\alpha\in(1,2)$, inequality (5.7), and therefore also (5.5), cannot hold for any $\varphi,\theta\in(-\pi,\pi)$. Thus, for whatever point $z$ on the boundary of $\mathscr D(\alpha)$ (that is, satisfying (5.4)), its image $f(z)$ cannot lie in the interior of $\mathscr W(\alpha)$.
In Figure 4 we present an illustration of the above arguments for $f'(1)=\alpha$. The point $z$ wanders along the boundary of $\mathscr{D}(\alpha)$, while the corresponding disc $\mathscr{U}(z)$ rolls about the boundary of $\mathscr{W}(\alpha)$. Now we fix some point $w\in\mathscr W(\alpha)$ and implement the change if trajectory mentioned above. We indicate the part of the disc $\mathbb D$ in which the modified part of the trajectory must lie. Set
$$
\begin{equation}
\varepsilon=\min\biggl(\sqrt{\frac{\alpha-1}\alpha},\, \frac{|1-w|}4\biggr)
\end{equation}
\tag{5.8}
$$
and consider the sector
$$
\begin{equation}
\biggl\{z\in\mathbb D\colon|1-z|<\varepsilon, \,|{\arg(1-z)}|<\arctan \frac 1{\sqrt{\alpha-1}}\biggr\}.
\end{equation}
\tag{5.9}
$$
Notice that all points on the curve (5.4) that occur in the $\varepsilon$-neighbourhood of the point $z=1$ lie in the sector (5.9). Fix some point $z$ in this sector. In view of (5.8) we have the estimate
$$
\begin{equation}
\begin{aligned} \, \notag (\alpha-1)\frac{|1-z|^2}{1-|z|^2} &=(\alpha-1)\frac{|1-z|}{2\cos(\arg(1-z))-|1-z|} <(\alpha-1)\frac{|1-z|}{2\sqrt{(\alpha-1)/\alpha}-\varepsilon} \\ &<\sqrt{\alpha(\alpha-1)}\,|1-z|<\varepsilon\sqrt{\alpha(\alpha-1)}. \end{aligned}
\end{equation}
\tag{5.10}
$$
Since $\mathscr U(z)$ can equivalently be expressed in the form
$$
\begin{equation*}
\biggl|\zeta-\frac z{p+1}\biggr|\leqslant \frac {p|z|}{p+1}, \qquad p= (\alpha-1)\frac{|1-z|^2}{1-|z|^2},
\end{equation*}
\notag
$$
taking (5.8)–(5.10) into account, for each $\zeta$ in the disc $\mathscr U(z)$ we deduce the estimate
$$
\begin{equation}
\begin{aligned} \, \notag |1-\zeta| &\leqslant |1-z|+\frac{p|z|}{p+1}+\biggl|\zeta-\frac{z}{p+1}\biggr| \leqslant |1-z|+\frac{2p|z|}{p+1}<|1-z|+2p \\ &<\varepsilon +2\varepsilon\sqrt{\alpha(\alpha-1)}<4\varepsilon<|1-w|. \end{aligned}
\end{equation}
\tag{5.11}
$$
We make a counterclockwise rotation around the point $w$ along the smooth curve formed by a part of the curve (5.4) completed by an arbitrary smooth curve lying in the sector (5.9) inside the $\varepsilon$-neighbourhood of $z=1$. Cutting a corner in the $\varepsilon$-neighbourhood of the point $z=1$ in this way is possible because all points on the curve (5.4) that occur in this neighbourhood lie inside the sector (5.9). In travelling along the curve (5.4) the image of $z$, as we have already noted, cannot occur in the domain $\mathscr W(\alpha)$ and thus cannot make a full rotation around $w$. On the other hand, as the point goes along the modified fragment in a neighbourhood of $z=1$, $f(z)$ cannot wind around $w$ because of (5.11). Thus, when $z$ completes a circuit, the argument of $f(z)-w$ increases by $2\pi$. Since the modified part of the curve can lie arbitrarily close to $z=1$, the equation $f(z)=w$ has a unique solution in the whole of $\mathscr D(\alpha)$. Since we have established the existence of a unique solution of the equation $f(z)=w$ for an arbitrary $w\in\mathscr W(\alpha)$, we have proved the first part of the theorem. Finally, to prove the last part of the theorem it suffices to observe that, as ${\alpha\to 1+0}$, the curve (5.3) is continuously deformed into the circle $\mathbb T$, while as $\alpha\to 2-0$, it is deformed into the line segment $[0,1]$. On the other hand, for $\theta\in [0,\pi)$ the function $f_{\theta}(z)=z \bigl(1-e^{i\theta}+2 e^{i\theta}z\bigr)/\bigl(2-(1-e^{i\theta})z\bigr)$ belongs to the class $\mathscr B_2[0, 1]$, and the inverse function of $f_{\theta}$ has the branch point $w_\theta=\tan^4(\theta/4)$. Hence each point in $\mathbb D$ is a branch point of the inverse of some function in $\mathscr B_2[0, 1]$. Thus, even the class $\mathscr B_2[0, 1]$ has no nonempty domains of univalent covering. The proof of Theorem 1 is complete. Remark 8. The curve with the property that the discs $\mathscr U(z)$ corresponding to its points are disjoint from $\mathscr W(\alpha)$ is unique. Such a curve must be the boundary of $\mathscr D(\alpha)$. If $f'(1)=\alpha$ and $z$ does not lie on the boundary of $\mathscr D(\alpha)$, then $\mathscr U(z)$ intersects the domain $\mathscr W(\alpha)$. Remark 9. Geometrically, we can characterize the boundary of the sharp univalent covering domain for the class $\mathscr B_{\alpha}[0, 1]$ as the envelope of the family of discs $\mathscr U(z)$ corresponding to the points $z$ on the boundary of the sharp univalence domain for the same class. Remark 10. We see from the proof of the theorem that its statement can be refined: the domain $\mathscr X$ onto which a branch of the inverse function of $f\in \mathscr B_{\alpha}[0, 1]$ maps conformally the domain $ \mathscr W(\alpha)$ can be taken inside $\mathscr D(\alpha)$. § 6. Maximal univalent covering discIn this section we find the disc of maximum radius that is univalently covered by all functions in the class $\mathscr B_{\alpha}[0, 1]$. Of course, its centre is not at the origin, but lies slightly to the right of it. Theorem 2. Let $f\in \mathscr B_{\alpha}[0, 1]$, $\alpha\in(1,2)$. Then there exists a function inverse to $f$ and mapping the disc where $k_1=\sqrt{\alpha-1}$, $k_2=\sqrt{\alpha+1}$ and $k_3=\sqrt{2\alpha-1}$, conformally onto a domain $\mathscr X\subset \mathbb D$. For whatever the disc $\mathscr O$ of radius greater than that of $\mathscr C(\alpha)$ is, there exists a function $f\in \mathscr B_{\alpha}[0, 1]$ without an inverse function in $\mathscr O$. Proof. We start with the proof of the second part of the theorem. It is sufficient to show that a disc of radius $R>R(\alpha)$, where
$$
\begin{equation*}
R(\alpha)=\frac{2\alpha(k_3-k_1 k_2)}{(k_2+k_1 k_3)^2},
\end{equation*}
\notag
$$
cannot be inscribed in $\mathscr W(\alpha)$ (see (5.2)).
As already mentioned, the boundary of $\mathscr W(\alpha)$ is the curve (5.3). Hence the $y$-coordinate of a point on this curve has the form
$$
\begin{equation*}
y=\frac{\alpha(2-\alpha)\sin \theta}{(\alpha+2k_1\cos({\theta}/{2}))^2}, \qquad \theta\in(-\pi,\pi].
\end{equation*}
\notag
$$
For $\theta$ ranging over $(0,\pi)$ the derivative $y'_{\theta}$ vanishes at the point $\theta =\theta_0$ such that $\cos(\theta_0/2)=(k_2k_3-k_1)/(2\alpha)$. Hence on the boundary of $\mathscr W(\alpha)$ the $y$-coordinate is greatest for $\theta=\theta_0$. Noting that $y(\theta_0)=R(\alpha)$, we conclude that $\mathscr W(\alpha)$ lies fully in a strip of length $2R(\alpha)$. However, one cannot inscribe a disc of radius $R>R(\alpha)$ in such a strip, still less in $\mathscr W(\alpha)$. We turn to the proof of the first part of the theorem. It is sufficient to verify that all boundary points of $\mathscr C(\alpha)$ lie in the closure of $\mathscr W(\alpha)$. Consider an arbitrary boundary point of $\mathscr C(\alpha)$,
$$
\begin{equation*}
w_{\varphi}=\frac{2k_1(k_2k_3-k_1)+2\alpha(k_3-k_1k_2)e^{i\varphi}} {(k_2+k_1k_3)^2}, \qquad \varphi\in(-\pi,\pi];
\end{equation*}
\notag
$$
let us see that it lies in the closure of the domain $\mathscr W(\alpha)$, that is,
$$
\begin{equation}
\frac {|1-w_{\varphi}|}{1-|w_{\varphi}|}\leqslant \frac{\alpha}{2k_1}.
\end{equation}
\tag{6.2}
$$
Substituting $w_{\varphi}$ into the left-hand side of (6.2), after some transformations we obtain
$$
\begin{equation}
\frac{|1-w_{\varphi}|}{1-|w_{\varphi}|}= \frac{2\alpha\sqrt{(k_2 k_3-k_1)^2-2\alpha(k_3-k_1k_2)\cos \varphi}}{(k_2+k_1k_3)^2-\sqrt{(k_2k_3-k_1)^4+8\alpha k_1(k_3-k_1k_2)(k_2k_3-k_1)\cos\varphi}}.
\end{equation}
\tag{6.3}
$$
Taking (6.3) into account we can write (6.2) as follows:
$$
\begin{equation}
\begin{aligned} \, \notag &4\sqrt{(\alpha-1)((k_2k_3-k_1)^2-2\alpha(k_3-k_1k_2)\cos\varphi)} \\ &\qquad \leqslant (k_2+k_1k_3)^2-\sqrt{(k_2k_3-k_1)^4+8\alpha k_1(k_3-k_1k_2)(k_2k_3-k_1)\cos\varphi}. \end{aligned}
\end{equation}
\tag{6.4}
$$
Since the cosine function is even, we need only establish (6.4) on $[0,\pi]$. For ${\varphi=\pi/2}$ inequality (6.4) turns to equality, so it is sufficient to verify the corresponding inequalities for the derivatives of the left- and right-hand sides of (6.4) on the intervals $[0,\pi/2)$ and $(\pi/2, \pi]$. Namely, it remains to see that for all $\varphi\in[0,\pi/2)$ we have
$$
\begin{equation}
\begin{aligned} \, &\frac{4\alpha k_1(k_3-k_1k_2)\sin\varphi}{\sqrt{(k_2k_3-k_1)^2-2\alpha(k_3-k_1k_2)\cos\varphi}} \nonumber \\ &\qquad <\frac{4\alpha k_1(k_3-k_1k_2)(k_2k_3-k_1)\sin\varphi}{\sqrt{(k_2k_3-k_1)^4+ 8\alpha k_1(k_3-k_1k_2)(k_2k_3-k_1)\cos\varphi}}, \end{aligned}
\end{equation}
\tag{6.5}
$$
while for all $\varphi\in(\pi/2,\pi]$ we have the reverse inequality. Squaring (6.5), we can transform it into $\cos \varphi > 0$, which holds for $\varphi\in[0,\pi/2)$, and we can transform the reverse inequality into $\cos \varphi < 0$, which holds for $\varphi\in(\pi/2,\pi]$. The proof is complete. Figure 5 shows the mutual position and sizes of the discs $\mathscr E(\alpha)$ and $\mathscr C(\alpha)$ (see (5.1) and (6.1)). Note that, as $\alpha\to 1+0$, the centre of $\mathscr C(\alpha)$ tends to zero, while the radii $r(\alpha)$ and $R(\alpha)$ of $\mathscr E(\alpha)$ and $\mathscr C(\alpha)$ have the asymptotic expressions
$$
\begin{equation*}
1-r(\alpha)\sim 4\sqrt{\alpha-1}\quad\text{and} \quad 1-R(\alpha)\sim \sqrt{2}\sqrt{\alpha-1}.
\end{equation*}
\notag
$$
Thus, for $\alpha$ close to $1$ both discs are close to $\mathbb D$ and their radii are different from $1$ by quantities of the same order. On the other hand, if $\alpha$ is close to 2, then the picture is different. As $\alpha\to 2-0$, the centre of $\mathscr C(\alpha)$ tends to $z=1/3$, while the radii $r(\alpha)$ and $R(\alpha)$ of the discs $\mathscr E(\alpha)$ and $\mathscr C(\alpha)$ have the asymptotic expressions
$$
\begin{equation*}
r(\alpha)\sim \frac{(2-\alpha)^2}{16}\quad\text{and} \quad R(\alpha)\sim \frac{2-\alpha}{3\sqrt{3}}.
\end{equation*}
\notag
$$
We see from the asymptotic relations that for $\alpha$ close to 2 the disc $\mathscr C(\alpha)$ has a much larger radius than $\mathscr E(\alpha)$.
§ 7. The case of an arbitrary interior fixed pointLet us transfer the result of Theorem 1 to the case of an arbitrary fixed point $q\in \mathbb D$. To do this we consider the linear fractional transformation
$$
\begin{equation*}
w=T(z)=\frac{1-\overline{q}}{1-q}\,\frac{z-q}{1-\overline{q}z}.
\end{equation*}
\notag
$$
It takes the unit disc $\mathbb D$ to itself and satisfies the conditions $T(q)=0$ and $T(1)=1$. Now if $f\in \mathscr B[0, 1]$, then the composition $\widetilde{f}(z)=T^{-1}\circ f\circ T(z)$ belongs to the class $\mathscr B[q, 1]$ and, moreover, $\widetilde{f}'(1)=f'(1)$. In other words, $T$ maps the class $\mathscr B_\alpha[q, 1]$ bijectively onto $\mathscr B_\alpha[0, 1]$. Applying $T$ to the domain $\mathscr W (\alpha)$ (see (5.2)) we obtain the following result. Theorem 3. Let $f\in \mathscr B_{\alpha}[q, 1]$, $\alpha\in(1,2)$. Then there exists a function inverse to $f$ and mapping the domain conformally onto some domain $\mathscr X\subset \mathbb D$. For whatever domain $\mathscr V$ such that $\mathscr W(\alpha, q)\subset \mathscr V \subset \mathbb D$, $\mathscr V\neq \mathscr W(\alpha, q)$, there exists a function $f\in \mathscr B_{\alpha}[q, 1]$ that has no inverse function in $\mathscr V$. Finally, for $\alpha\geqslant 2$ the class $\mathscr B_{\alpha}[q, 1]$ has no nonempty domain of univalent covering. Remark 11. It is easy to verify that the boundary of the domain (7.1) is the curve $w_{q}(\theta)$, $\theta\in(-\pi,\pi]$, defined by the equation In Figure 6 we show the domain $\mathscr W(\alpha, q)$ with an interior fixed point $q$ in general position. Note that, of course, the position of the interior fixed point affects the size and shape of the domain $\mathscr W(\alpha, q)$. On the other hand the opening of the corner of $\mathscr W(\alpha, q)$ at the boundary fixed point is independent of the position of the interior fixed point. The size of the opening depends exclusively at the value of the derivative at the boundary fixed point: it is $2\arctan\bigl((1-\alpha/2)/\sqrt{\alpha-1}\bigr)$. In conclusion we use the above results to show that there is no analogue of Theorem D for the class $\mathscr B_{\alpha}\{1\}$, $\alpha>1$. Namely, the class $\mathscr B_{\alpha}\{1\}$ is too wide for indicating a common domain of univalent covering. Moreover, this problem has no nontrivial solution even for the subclass $\mathscr B'_{\alpha}\{1\}$ of $\mathscr B_{\alpha}\{1\}$ consisting of functions with an attracting fixed point in the disc $\mathbb D$. Theorem 4. For no $\alpha>1$ does there exist a nonempty domain of univalent covering for the class $\mathscr B'_{\alpha}\{1\}$. Proof. In the case when $\alpha\geqslant 2$ the required result follows immediately from Theorem 3. Consider the case when $\alpha\in(1,2)$ in detail. Since $\mathscr B'_{\alpha}\{1\}=\bigcup_{q\in\mathbb D}\mathscr B_{\alpha}[q,1]$, a domain of univalent covering for $\mathscr B'_{\alpha}\{1\}$ must be a domain of univalent covering for each class $\mathscr B_{\alpha}[q,1]$, $q\in\mathbb D$. However, then by Theorem 3 the univalent covering domain for $\mathscr B'_{\alpha}\{1\}$ must lie in $\mathscr W(\alpha,q)$ for each $q\in\mathbb D$.
On the other hand, for points on the curve $w_{q}(\theta)$, $\theta\in (-\pi,\pi)$, which bounds the domain $\mathscr W(\alpha,q)$ (see Remark 11) we have the estimate
$$
\begin{equation*}
\begin{aligned} \, &|1-w_{q}(\theta)| \\ &=\frac{2\alpha\cos(\theta/2)|1-q|^2}{|(\alpha-1)(1- \overline{q}-\overline{q}(1-q)e^{i \theta})+(1-\overline{q})e^{i \theta}-\overline{q}(1-q)+2\sqrt{\alpha-1}\, e^{i\theta /2}(1-|q|^2)|} \\ &\leqslant \frac{2\alpha\cos(\theta/2)|1-q|^2}{\operatorname{Re}(e^{-i\theta /2}(\alpha-1+|q|^2-\alpha\overline{q})+e^{i\theta /2}((\alpha-1)|q|^2+1-\alpha\overline{q})\,{+}\,2\sqrt{\alpha-1}\, (1\,{-}\,|q|^2))} \\ &\leqslant \frac{\alpha\cos(\theta/2)|1-q|^2}{\sqrt{\alpha-1}\, (1-|q|^2)}, \end{aligned}
\end{equation*}
\notag
$$
because It follows from this estimate that $\angle \lim_{q\to 1} w_{q}(\theta)=1$ uniformly in $\theta\in (-\pi,\pi)$. Thus, $\bigcap_{q\in \mathbb D}\mathscr {W}(\alpha,q) =\varnothing$. The proof is complete. 
Citation in format AMSBIB
\Bibitem{KudSol24} |
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