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This article is cited in 3 scientific papers (total in 3 papers) Extremal problems in geometric function theory F. G. Avkhadieva, I. R. Kayumovab, S. R. Nasyrova a Kazan' (Volga Region) Federal University b Saint Petersburg State University
Russian version:
Uspekhi Matematicheskikh Nauk, 2023, Volume 78, Issue 2(470), DOI: https://doi.org/10.4213/rm10076 
Bibliographic databases:
    1. IntroductionThis paper is devoted to applications of conformal mappings to various problems in complex analysis. It should be mentioned that this area is so wide that we cannot present all results obtained to date. Here we describe some achievements connected with our own results and falling under the above topics. Simple and very important examples of conformal isomorphisms of plane domains are given by linear fractional automorphisms of the extended complex plane (Riemann sphere). The fruitful use of conformal automorphisms in mathematics found its starting point in the proof of Riemann’s theorem on conformal mappings, which claims that any simply connected domains in the plane with more than one boundary point can be mapped conformally onto a disc. Thus, conformal automorphisms of a disc appear quite often in the solutions of various problems in mathematics and mechanics, including applied problems. Our survey consists of several parts. In § 2 we describe results on integral means of conformal mappings, which underlie the solution of many difficult problems which remained open since the beginning of the 20th century. For example, we can mention Makarov’s results on the metric properties of the harmonic measure on Jordan curves. Section 3 is concerned with inequalities of Schwarz–Pick and Bohr type. The efficacy of the Schwarz–Pick inequalities has been shown by several generations of researchers. Bohr’s classical inequality is shockingly elegant and can also be established using the Schwarz–Pick inequalities. Most extremal functions there are related to conformal automorphisms of a disc. However, as Bombieri and Bourgain showed, in some cases extremals for a sum of Bohr type can be functions with properties close to Kahane’s ultraflat polynomials. Sections 4–6 are concerned with analytic and geometric problems related to integral inequalities of Hardy and Rellich type in domains of arbitrary shape. In particular, we present well-known criteria due to Maz’ya and Ancona for the positivity of constants in basic Hardy-type inequalities. Avkhadiev’s recent results showing that the boundary of a plane domain being perfect is a criterion for the positivity of constants in a number of inequalities of Hardy and Rellich type. Two-sided estimates for constants in terms of the Euclidean maximum modulus of the domain are also presented. Sections 8–10 concern some applications of conformal and generalized conformal moduli to problems in geometric function theory. For example, we examine Vuorinen’s problem on the asymptotic behaviour of the conformal modulus of doubly connected domains dilated along the $x$-axis with coefficient tending to infinity. On the basis of Ahlfors’s and Warshawski’s classical theorems we also establish a formula for the asymptotic behaviour of the conformal modulus of a quadriateral under dilations. In addition, we obtain formulae for the variation of Robin’s capacity under variations of the boundary of the domain and present their applications to aero- and hydromechanics and, in particular, to the solution of the generalized Lavrentiev problem. Next we consider one-parameter families of rational and elliptic functions and present a description of such families in terms of differential equations. We find systems of differential equations for critical points and poles of functions (and for their moduli, in the case of elliptic functions) under the assumption that their critical values vary in accordance with a prescribed law. The results obtained are used to investigate the conformal moduli of the domains equal to the exterior of two line segments; we establish some results on the monotonicity of the conformal modulus under the motion of one segment along the straight line containing it so that its length is constant and the second segment is fixed. In the last section we present an overview of the results on Smale’s well-known problem of critical values of polynomials. The problem itself is purely algebraic; it can be solved by algebraic methods, and it is easy to show that the answer is an algebraic number. However, in actual fact even the case $n=4$ is quite non-trivial (without using machine calculations), while for $n \geqslant 10$ even computers are of little help (unless you construct an explicit example by chance). It turns out that state-of-the-art methods in geometric function theory and, in particular, the symmetrization method developed intensively by Dubinin [57] can be used in the solution of problems in this area. It should be admitted that our survey covers only the part of investigations in geometric function theory in whose development these authors themselves have actively been involved. In particular, we do not include interesting and important results due to Goryainov [75], [76], Prokhorov [131], [132], and Starkov [142], [143]. 2. Integral means of conformal mappingsEstimates for integral means of conformal mappings are very often used in the solution of various problems in the geometric theory of functions of a complex variable. We can mention here the area theorem proved by Gronwall (for instance, see [74] or [102]), which was the starting point for the study of extremal problems in the class of univalent functions. Also note that, by Cauchy’s integral formula the problem of estimates for coefficients of univalent functions is in fact connected with estimates for integral means. One of the main problems in geometric function theory in the 20th century, which was a guiding star for many efficient methods for the solution of extremal problems, was Bieberbach’s conjecture that $|a_n| \leqslant n$, where the $a_n$ are the Taylor coefficients of a function in the class $S$. Recall that the class $S$ consists of univalent holomorphic functions $f$ in the disc $\mathbb{D}=\{z\colon|z|< 1\}$ such that $f'(0)-1=f(0)=0$. Before de Branges proved the Bieberbach conjecture in 1985, problems of integral means for the modulus of a univalent function (or the modulus of its derivative) had been viewed as an auxiliary tool in estimates for coefficients in $S$. This problem was considered by a number of well-known authors, including Littlewood, Goluzin, Lebedev, Milin, and others. Makarov [104] and Carleson and Jones [39] revealed the non-trivial links existing between integral means and the boundary behaviour of conformal mappings. Their results attracted considerable attention of experts to problems of estimates for integral means in investigations in geometric function theory. Let $\Omega$ be a bounded simply connected plane domain and $f$ be a conformal mapping of the disc $\mathbb{D}$ onto $\Omega$. By the area theorem
$$
\begin{equation*}
\iint_{\mathbb{D}}|f'(re^{i t})|^2 r\,dr\,dt \leqslant \pi M^2, \qquad M=\sup_{|z|<1}|f(z)|,
\end{equation*}
\notag
$$
which yields directly
$$
\begin{equation*}
\int_{0}^{2\pi}|f'(re^{it})|^2\,dt=O\biggl(\frac{1}{1-r}\biggr), \qquad r \to 1.
\end{equation*}
\notag
$$
Hence we see that in integrating along level curves the growth order of the modulus of the derivative of a univalent function reduces by one for $p \geqslant 2$:
$$
\begin{equation*}
\int_{0}^{2\pi}|f'(re^{i\theta})|^p\,dt= O\biggl(\frac{1}{(1-r)^{p-1}}\biggr), \qquad r \to 1.
\end{equation*}
\notag
$$
Thus it is quite natural to anticipate that for each fixed $p>1$ we have the same relation. It is obvious that for $p=1$ such a behaviour is not always possible, as there are bounded simply connected domains with non-rectifiable boundaries. Thus, the geometry of the domain is essential for $p=1$. It turned out subsequently that the geometry of the domain is also significant for all $p<2$. To investigate the behaviour of integral means, it proves to be a good idea to look at the so-called spectrum of integral means:
$$
\begin{equation*}
\beta_f(p)=\limsup_{r \to 1} \frac{\log\int_0^{2\pi}|f'(re^{i\theta})|^p\,d\theta}{|\log(1-r)|}\,,
\end{equation*}
\notag
$$
which is in fact the growth order of the integral means of the derivative. For ‘nice’ domains (for example, domains with bounded boundary rotation) $\beta_f(p)$ is a piecewise linear function of $p$. We note three extremely important results on the spectrum of integral means. 1) Makarov [104] showed that if the set $A\subset \partial\mathbb{D}$ is Borel measurable and the conformal mapping extends continuously to the boundary of $\mathbb{D}$, then for each $q>0$
$$
\begin{equation*}
\dim f(A) \geqslant \frac{q \dim A}{\beta_f(-q)+q+1-\dim A}\,,
\end{equation*}
\notag
$$
where $\dim A$ is the Hausdorff dimension of $A$. 2) Carleson and Jones [39] showed that
$$
\begin{equation*}
\sup_{f \in S_1}\beta_f(1)=\alpha:=\sup_{f \in S_1} \limsup_{n \to \infty} \frac{\log|na_n|}{\log n}\,,
\end{equation*}
\notag
$$
where $S_1$ is the class of bounded univalent functions in $\mathbb{D}$ and the $a_n$ are the coefficients of the Taylor expansion of $f$. Note that the inequality $\sup\beta_f(1) \geqslant \alpha$ is quite easy to prove: this is based on the fact that the integral of the modulus of a function is not less than the modulus of the integral of this function. On the other hand the reverse inequality is highly non-trivial. 3) Pommerenke ([129], 241) established the following result. If $f(\mathbb{D})$ is a domain of John class (that is $f(\mathbb{D})$ has, in fact, no zero interior angles), then where $p$ is the unique solution of the equation $\beta_f(p)=p-1$ and $\operatorname{mdim}\partial f(\mathbb{D})$ is the upper Minkowski metric dimension of the set $\partial f(\mathbb{D})$.We must particularly point out Makarov’s results [105] on the use of probabilistic methods in geometric function theory. A great number of complicated and non-trivial problems in complex analysis were solved with their help. The reader can also find information on all these results in the remarkable monograph [73] by Garnett and Marshall. These results explain the complicated behaviour of $\beta_f(p)$. Carathéodory’s classical theorem states that a conformal mapping between domains with Jordan boundaries can be extended to a homeomorphism of the bounded domains, but it gives no information on how the linear measures of Borel subsets of the boundaries of these domains are distorted. An analysis of the behaviour of integral means allows one to shed light on this question. The reader can read more on the properties of the spectrum of integral means in [79]. One intriguing and not yet solved problem is Brannan’s conjecture, which can be stated as follows [36]. Let $\Omega$ be a simply connected plane domain with more than two boundary points, and let $\varphi$ be a conformal mapping of $\Omega$ onto the unit disc $\mathbb{D}$. Brennan conjectured that $\varphi' \in L_p(\Omega)$ for $4/3 < p < 4$, that is,
$$
\begin{equation*}
\int_\Omega |\varphi'|^p\,dx\,dy < \infty \quad\text{for } \frac{4}{3} < p < 4.
\end{equation*}
\notag
$$
The question of integral means is not limited to conformal mappings. Problems similar to Brennan’s conjecture also arise for rational functions, which are in fact multivalent mappings. Consider a problem of this type connected with estimates for integrals of bounded rational functions, which was originally considered by Dolzhenko [53] for domains with sufficiently smooth boundaries. Let $G$ be an arbitrary domain with bounded area $S$, $R$ be a rational function of order $n$, and let $p \geqslant 1$. If $|R| \leqslant 1$ in $G$, then it follows from Hölder’s inequality that
$$
\begin{equation*}
\int_{G}|R'(z)|^p\,dx\,dy \leqslant \biggl(\int_{G}|R'(z)|^2\,dx\,dy\biggr)^{p/2}S^{1-p/2} \leqslant (\pi n)^{p/2}S^{1-p/2},
\end{equation*}
\notag
$$
and this bound is sharp in order, provided that we impose no additional constraints on the domain (for example, we do not assume it to have a rectifiable boundary). It turns out that in some cases this bound can be improved. By a curve of type K we mean a closed Jordan curve with continuous curvature $k(s)$ satisfying Hölder’s condition (here $s$ is the natural parameter). Let $G$ be a finitely connected domain with boundary curves of type K. Let $1\leqslant p \leqslant 2$, and let $R$ be a rational function of degree at most $n$ with poles outside $\overline{G}$. Dolzhenko ([53], Theorem 2.2) showed that there exists a constant $C$ depending on the domain $G$ and $p$ such that
$$
\begin{equation}
\int_{G}|R'(z)|^p\, dx\,dy \leqslant C n^{p-1}\sup_{z \in G}|R(z)|, \qquad p \in (1,2],
\end{equation}
\tag{1}
$$
while under the additional condition that $G$ is bounded,
$$
\begin{equation*}
\int_{G}|R'(z)|\,dx\,dy \leqslant C \log (n+1) \sup_{z \in G} |R(z)|.
\end{equation*}
\notag
$$
Subsequenty, integral inequalities for the derivatives of rational functions were considered by Peller [127], Semmes [136], Pekarskii [125], [126], Danchenko [44], and many others. Baranov and Kayumov [28] established (1) under significantly weaker assumptions about the domain, namely, that it has no zero interior angles. In addition, in [29] they were able to improve this inequality to the sharp (in order) bound
$$
\begin{equation}
\int_{G}|R'(z)|\,dx\,dy \leqslant C\sqrt{\log (n+1)}\,\sup_{z \in \Omega}|R(z)|.
\end{equation}
\tag{2}
$$
The fact that (2) is sharp in order is quite non-trivial; it relies on the repeated logarithm law for conformal mappings proved by Makarov in 1985 г. That result reads as follows. Let $f$ be a holomorphic function in the disc $\mathbb{D}$. Makarov [103] showed that there exists a positive constant $C$ such that
$$
\begin{equation}
\limsup_{r \to 1-}\frac{|f(r \zeta)|}{\sqrt{|\log(1-r)| \log\log|\log(1-r)|}} \leqslant C \|f\|_{\rm B}
\end{equation}
\tag{3}
$$
for almost all $\zeta$ on the circle $|\zeta|=1$, where is the standard Bloch seminorm. The reader can get updated from [89] on the investigations in the law of repeated logarithm in the Bloch class. Przytycki and Makarov conjectured that the Bloch seminorm (4) can be replaced in (3) by asymptotic dispersion, by analogy with the classical repeated logarithm law due to Kolmogorov and Khintchine. However, this conjecture failed: the corresponding counterexample is due to Bañuelos and Moore [27]. Note that asymptotic dispersion is important for contemporary investigations in geometric function theory (particularly, in the part where it overlaps with probability theory). The state-of-the-art situation in this area was described by Hendalmalm [78]. In [28] the example given by Bañuelos and Moore was adapted to showing that (2) is sharp; to do this, from the Bloch space the authors had to descend to the subspace of bounded functions. Recently Baranov and Kayumov were able to drop the assumption of the absence of zero interior angles in justifying estimates (1) and (2). The only condition remaining—which is quite a reasonable one—is that the domain must have a rectifiable boundary. By the way, this can also be relaxed and replaced by a restriction on the fractal dimension of the boundary. Of course, inequalities (1) and (2) will also be weaker in this case. Note that conformal mappings of simply connected domains onto a disc play a significant role in all these results. Also note that the multiply connected case can easily be reduced to the simply connected one by drawing a finite number of smooth cuts. 3. Schwarz–Pick and Bohr-type inequalitiesA simple but quite efficient example of how conformal isomorphisms can be used is the Schwarz–Pick inequalities. The basic inequality can be formulated as follows. Let $f$ be a holomorphic functions in the disc $\mathbb{D}$, and let $f(\mathbb{D}) \subset \mathbb{D}$. Then the following sharp inequality holds for each $z_0 \in \mathbb{D}$:
$$
\begin{equation*}
\biggl|\frac{f(z)-f(z_0)}{1-\overline{f(z_0)}f(z)}\biggr| \leqslant \biggl|\frac{z-z_0}{1-\overline{z}_0 z}\biggr|.
\end{equation*}
\notag
$$
The proof is easy to obtain using conformal automorphisms of discs in the $z$-plane and target plane, with a reference to Schwarz’s lemma. Dividing both sides by $|z-z_0|$ and letting $z \to z_0$ we obtain a well-known corollary to the Schwarz–Pick inequality:
$$
\begin{equation*}
|f'(z_0)| \leqslant \frac{1-|f(z_0)|^2}{1-|z_0|^2}\,.
\end{equation*}
\notag
$$
In particular, setting $z_0=0$ we obtain Here the $a_k$ are the coefficients of the Maclaurin series of $f$. Inequality (5) has a simple useful consequence: These inequalities can be obtained from (5) by symmetrizing $f$:
$$
\begin{equation*}
f_n(z):=\frac{1}{n} \sum_{k=0}^{n-1} f(e^{i 2k\pi/n} z).
\end{equation*}
\notag
$$
Note the most important role played by conformal automorphisms of the disc $(z-a)/(1-\overline{a}z)$ in extremal problems in the class of bounded holomorphic functions in $\mathbb{D}$, and the role of finite Blaschke products There are many results on Schwarz–Pick inequalities. The reader can find more information in the monograph [25] by Avkhadiev and Wirths and in the recent monograph [72]. To state the Schwarz–Pick inequality in full generality we need several definitions. Let $\Omega$ be a domain with at least three boundary points in the extended complex plane. By the Riemann–Poincaré theorem there exists a locally conformal orientation-preserving mapping $f\colon\mathbb{D}\to \Omega$ such that $\Omega=f(\mathbb{D})$, and in a sufficiently small neighbourhood of each point $z_0\in \Omega$ an element of the inverse function $F(z)=f^{-1}(z)$ is defined which can be continued analytically along any path in $\Omega$. All the values of all possible analytic extensions of this element to $\Omega$ lie in $\mathbb{D}$. This $f\colon\mathbb{D}\to \Omega$ is called a universal covering map. If $\Omega$ is simply connected, then the covering map $f\colon\mathbb{D}\to \Omega$ coincides with the univalent conformal mapping of the unit disc onto $\Omega$. On the other hand, if $\Omega$ is not simply connected, then the covering map is not injective. Let $T$ be a conformal automorphism of $\mathbb{D}$. Then the compositions of the form $f\circ T$ exhaust all covering map of $\mathbb{D}$ onto $\Omega$. It is well known that the Poincaré metric with Gaussian curvature $\kappa=-4$, which specifies the hyperbolic geometry in $\Omega$, has the coefficient
$$
\begin{equation*}
\lambda_\Omega(z)=\frac{1}{|f'(\zeta)|(1-|\zeta|^2)}\,,\qquad \zeta \in \mathbb{D},\quad z=f(\zeta)
\end{equation*}
\notag
$$
(see [4] and [74]). By the conformal invariance of the Poincaré metric in the unit disc this formula defines consistently the coefficient of the hyperbolic metric in a multiply connected domain, in spite of the fact that the covering mapping is not uniquely defined. The function $1/\lambda_\Omega(z)$, denoted by $R(z,\Omega)$, is called the hyperbolic radius. Thus, the hyperbolic radius is defined by the formula where $f\colon\mathbb{D}\to \Omega$ is a locally univalent covering map of the unit disc onto the domain $\Omega \subset \overline{\mathbb C}$ with at least there boundary points.The hyperbolic radius has the following basic properties: (a) $R(\,\cdot\,,\Omega)\colon \Omega \to (0,\infty]$ is a real analytic function at all finite points in the domain; (b) if $\infty \in \Omega$, then the following functions are real analytic in a small neighbourhood of the point at infinity:
$$
\begin{equation*}
\frac{R(z,\Omega)}{|z|^2}\,, \quad \frac{|\nabla R(z,\Omega)|}{|z|}\,, \quad\text{and}\quad \Delta R(z,\Omega).
\end{equation*}
\notag
$$
Furthermore, the limits
$$
\begin{equation*}
\lim_{z\to \infty}\frac{R(z,\Omega)}{|z|^2}>0, \quad \lim_{z\to \infty}\frac{|\nabla R(z,\Omega)|}{|z|}>0, \quad\text{and}\quad \lim_{z\to \infty}\Delta R(z,\Omega)=4
\end{equation*}
\notag
$$
exist; (c) the formula $|f'(\zeta)|(1-|\zeta|^2)=R(f(\zeta),\Omega)$, implies the following equivalent forms of Liouville’s equation:
$$
\begin{equation*}
\Delta \log R=-\frac{4}{R^2}\quad\text{and} \quad R \Delta R=|\nabla R|^2-4,
\end{equation*}
\notag
$$
where $R=R(z,\Omega)$, $z=x+iy \in \Omega$. Recall that $R(z,\mathbb{D})=1-|z|^2$, $z\in \mathbb{D}$. Let $A(\Omega,\Pi)$ denote the set of locally holomorphic or meromorphic (in general, multivalued) functions $F\colon\Omega \to \Pi$ in the domain $\Omega \subset \overline{\mathbb{C}}$ such that all possible values of $F(z)$ lie in the domain $\Pi \subset \overline{\mathbb{C}}$. In full generality the Schwarz–Pick inequality can be stated as follows. Theorem 3.1. Let $\Omega\subset\overline{\mathbb{C}}$ and $\Pi\subset\overline{\mathbb{C}}$ be two domains of hyperbolic type. If then at each point $z\in \Omega$. If $z\ne \infty$ and $F(z)\ne \infty$, then if and only if $F$ is a locally conformal mapping of $\Omega$ onto $\Pi$. Several generalizations of the Schwarz–Pick inequality are known. We present only three results, due to Avkhadiev and Wirths, on the extension of this inequality to higher derivatives of order $n\geqslant 2$. Following [25], for derivatives of an arbitrary functions $F \in A(\Omega,\Pi)$ consider the inequality
$$
\begin{equation*}
\frac{1}{n!}|F^{(n)}(z)| \leqslant C_n(\Omega,\Pi)\, \frac{R(F(z),\Pi)}{R^n(z,\Omega)}\,,
\end{equation*}
\notag
$$
where $C_n(\Omega,\Pi)$ is the least constant possible here, that is, the constant defined by
$$
\begin{equation*}
C_n(\Omega,\Pi)=\sup_{z\in \Omega}\,\sup_{F \in A(\Omega,\Pi)}\, \frac{R^n(z,\Omega)|F^{(n)}(z)|}{n!R(F(z),\Pi)}\,.
\end{equation*}
\notag
$$
By the Schwarz–Pick inequality it is obvious that $C_1(\Omega,\Pi)=1$ for any two domains of hyperbolic type $\Omega\subset\overline{\mathbb{C}}$ and $\Pi\subset\overline{\mathbb{C}}$, while for $n\geqslant 2$ we can a priori say only that $0< C_n(\Omega, \Pi) \leqslant\infty$. The following result holds. Theorem 3.2 ([25]). Let $\Omega\subset{\mathbb{C}}$ and $\Pi\subset\mathbb{C}$ be convex domains such that $\Omega\ne\mathbb{C}$ and $\Pi\ne\mathbb{C}$. Then for each $n\geqslant 2$, at each point $z\in \Omega$, for any function $F\in A(\Omega,\Pi)$ where $2^{n-1}$ is the sharp constant, namely, $C_n(\Omega,\Pi)=2^{n-1}$ for any pair of convex domains $\Omega\subset\mathbb{C}$ and $\Pi\subset{\mathbb{C}}$ such that $\Omega\ne\mathbb{C}$ and $\Pi\ne\mathbb{C}$. The reader can find several analogues of this theorem in [25], which are related to various geometric assumptions about $\Omega$ and $\Pi$. In particular, if $\Omega\subset\mathbb{C}$ and $\Pi\subset\mathbb{C}$ are simply connected domains of hyperbolic type, then $C_n(\Omega,\Pi)\leqslant 4^{n-1}$. We can only find $C_n(\Omega,\Pi)$ for arbitrary domains $\Omega\subset\mathbb{C}$ and $\Pi\subset\mathbb{C}$ of hyperbolic type in the case when $n=2$. Namely, Avkhadiev and Wirths ([25], Ch. 4) established the following theorem. Theorem 3.3. The following equality holds for arbitrary domains $\Omega\subset\mathbb{C}$ and $\Pi\subset\mathbb{C}$ of hyperbolic type: The condition $\sup_{z\in \Omega}|\nabla R(z,\Omega)|< \infty$ is a criterion for the uniform perfectness of the boundary of the domain $\Omega\subset{\mathbb{C}}$ of hyperbolic type (for instance, see [18] or [25]). Hence Theorem 3.3 has the following consequence. Corollary 3.3.1. For two domains of hyperbolic type $\Omega\subset{\mathbb{C}}$ and $\Pi\subset{\mathbb{C}}$, if and only if the boundaries of both $\Omega$ and $\Pi$ are uniformly perfect sets. We return to domains $\Omega\subset{\mathbb{C}}$ with uniformly perfect boundaries and their geometric characterisations in terms of the maximum moduli of domains in § 6, when we describe integral inequalities of Hardy and Rellich type. If one of the domains $\Omega\subset\overline{\mathbb{C}}$ and $\Pi\subset\overline{\mathbb{C}}$ contains a point at infinity, then $C_2(\Omega,\Pi)=C_2(\Pi,\Omega)=\infty$ by Theorem 3.3, because in a neighbourhood of $z = \infty$ the modulus of the gradient of the hyperbolic radius has the order $O(|z|)$. In these circumstances it makes sense to consider inequalities of the form
$$
\begin{equation*}
\frac{1}{n!}\,|F^{(n)}(z)| \leqslant M_n(z,\Omega,\Pi)\, \frac{R(F(z),\Pi)}{R^n(z,\Omega)}\,,
\end{equation*}
\notag
$$
where $M_n(z,\Omega,\Pi)$ is the least constant possible here< that is,
$$
\begin{equation*}
M_n(z,\Omega,\Pi)=\sup_{F \in A(\Omega,\Pi)}\, \frac{R^n(z,\Omega)|F^{(n)}(z)|}{n!\,R(F(z),\Pi)}\,.
\end{equation*}
\notag
$$
Clearly, by the Schwarz–Pick inequality $M_1(z,\Omega,\Pi)\equiv1$ for any two domains of hyperbolic type $\Omega\subset\overline{\mathbb{C}}$ and $\Pi\subset\overline{\mathbb{C}}$. It turns out that for $n\geqslant 2$ the quantity $M_n(z,\Omega,\Pi)$ depends essentially on the point $z\in \Omega$ and the properties of $\Omega$ and $\Pi$. To state the following theorem we require the hyperbolic distances $\rho_{\Omega}(z,\infty)$ and $\rho_{\Pi}(F(z),\infty)$. Recall that the hyperbolic distance $\rho_\mathbb{D}(0,p)$ between the points $0$ and $p\in (0,1)$ in the unit disc $\mathbb{D}$ is defined by
$$
\begin{equation*}
\rho_\mathbb{D}(0,p)=\frac{1}{2}\log\frac{1+p}{1-p} \quad\Longleftrightarrow\quad p=\tanh\rho_\mathbb{D}(0,p),
\end{equation*}
\notag
$$
where, as usual, $\tanh x= (e^x-e^{-x})/(e^x+e^{-x})$. Theorem 3.4 ([25]). Let $\Omega\subset\overline{\mathbb{C}}$ and $\Pi\subset\overline{\mathbb{C}}$ be simply connected domains of hyperbolic type. Let $\infty \in \Omega$ and $\infty \in \Pi$. Then for each $n\geqslant 2$, at each point $z\in \Omega$, for any $F\in A(\Omega,\Pi)$ The reader can find a number of related results in [23] and [24]. Let $\mathcal B$ denote the class of holomorphic functions $f$ in the unit disc $\mathbb{D}$ such that $|f(z)|<1$ for all $z\in \mathbb{D}$. We expand a function $f$ in the Taylor series: $f(z)=\sum_{k=0}^{\infty} a_kz^k$. Bohr’s classical result ensures that there exists a positive constant $\rho$ such that
$$
\begin{equation}
M^f(r):=\sum_{k=0}^{\infty}|a_k|r^k \leqslant 1 \quad\text{for all } r=|z|\leqslant \rho,
\end{equation}
\tag{8}
$$
and the value $\rho=1/3$ is optimal here; it is called the Bohr radius [33]. Note that, originally, Bohr proved (8) for $\rho=1/6$. Subsequently, Wiener, M. Riesz, and Schur proved the sharp inequality $r=|z|\leqslant 1/3$ independently of one another. It is worth noting that we can also consider the Bohr radius $\rho(f)$ for an arbitrary fixed function $f$, rather than for the whole class of functions satisfying $|f(z)|<1$ for $z\in \mathbb{D}$. (Moreover, we can define the Bohr radius for other classes of analytic functions in a similar way.) Then for each fixed function the Bohr radius is strictly greater than $1/3$. In the framework of the classical approach, when we regard the problem of finding the Bohr radius as an extremal one, there exists no extremal function. However, the quantity $1/3$ cannot be increased because there exists a sequence $(f_n)$ of conformal automorphisms of the disc such that $\rho(f_n)\to 1/3$; this sequence converges locally uniformly to a constant of modulus one. It is important to note that the Bohr radius for the whole class coincides with the Bohr radius for its subclass of conformal automorphisms of the disc. The above result was the starting point for extensive investigations of extremal problems induced by new Bohr-type inequalities in various classes of analytic functions. This is a good point to mention the excellent survey [1], the papers [124], [37], and [51], and the recent survey on Dirichlet series [134]. Note that inequalities (6) play a significant role in Bohr-type inequalities; they often underlie the proofs of inequalities in this area. In 2000 Djakov and Ramanujan [52] considered the Bohr phenomenon in a generalized setting. For $f\in {\mathcal B}$ and fixed $p>0$ consider the Bohr power sums $M_p^f(r)$, defined by Note that for $p=1$ the quantities $M_p^f(r)$ become the classical Bohr sums $M^f(r)$. The greatest constant $\rho_p$ such that
$$
\begin{equation*}
M_p^f(r)\leqslant 1 \quad\text{for all } r\leqslant \rho_p,
\end{equation*}
\notag
$$
is called the generalized Bohr radius for the class ${\mathcal B}$. Set and
$$
\begin{equation*}
r_p:=\sup\biggl\{r\colon a^p+\frac{r(1-a^2)^p}{1-r a^p} \leqslant 1, \ 0 \leqslant a<1\biggr\}=\inf_{a \in [0,1)} \frac{1-a^p}{a^p(1-a^{p})+(1-a^2)^p}\,.
\end{equation*}
\notag
$$
It is easy to note that $r_p$ is the generalized Bohr radius for the class of conformal automorphisms of the disc. Their role in this problem is decisive. Djakov and Ramanujan [52] showed that for fixed $p \in (1,2]$ and any function $f(z)=\sum_{k=0}^{\infty}a_kz^k$ in $\mathcal B$ we have $M_p^f (r)\leqslant 1$ for $r \leqslant T_p$, where and
$$
\begin{equation*}
m_p:=\frac{p}{(2^{1/(2-p)}+p^{1/(2-p)})^{2-p}}\quad \text{for}\ 1<p<2
\end{equation*}
\notag
$$
and $m_2:=\lim_{p \to 2} m_p=1$. In addition, for each $p \in (0,2)$ Aizenberg, Grossman, and Korobeinik [5] established the bound
$$
\begin{equation*}
M_p(r) \leqslant \max_{a \in [0,1]} \biggl[a^p+\frac{r\,2^p(1-a)^p}{1-r}\biggr].
\end{equation*}
\notag
$$
Kresin and Maz’ya [96] proved the following sharp inequality:
$$
\begin{equation*}
\sum_{n=m}^\infty|a_k|^p r^{k} \leqslant \frac{2^p r^m}{1-r} \biggl(\,\sup_{|\zeta|<1}\operatorname{Re}(f(\zeta)-f(0))\biggr)^p
\end{equation*}
\notag
$$
for $p \in (0,\infty)$ and $m \geqslant 1$. This interesting result and a number of others can also be found in their monograph [97]. Since the bounds found by Djakov and Ramanujan are not sharp, it is natural to ask about the optimal constant $\rho_p$ for $p\in (1,2)$. Is it true that the minimization of the Bohr radius over the class of bounded analytic function is equivalent to the same extremal problem in the class of conformal automorphisms of the disc, that is, does the equality $\rho_p=r_p$ hold? An affirmative answer was given by Kayumov and Ponnusami [94], who proved the following. Theorem 3.5. If $f(z)=\sum_{k=0}^{\infty} a_kz^k$ is a function in the class $\mathcal B$ and $0<p \leqslant 2$, then Moreover, This result generalizes an inequality proved by Bombieri [34] for $p=1$. Note that $M_p(r)=1$ for $p \geqslant 2$ and $r < 1$. In addition, $M_p(r)>1$ for all $p \in (0,1)$. Thus, the most interesting case is when $p \in [1,2)$. The following question is natural: is the bound for $M_p(r)$ in Theorem 3.5 close to sharp for $r$ close to one? For an answer we use an inequality due to Bombieri and Bourgain [35]. They showed that for each $\varepsilon >0$ there exists a positive constant $C(\varepsilon) $ such that
$$
\begin{equation*}
M_1(\rho) \geqslant \frac{1}{\sqrt{1-\rho^2}}- C(\varepsilon)\biggl(\log \frac{1}{1-\rho}\biggr)^{3/2+\varepsilon}\quad\text{for } \rho \geqslant \frac{1}{\sqrt{2}}\,.
\end{equation*}
\notag
$$
It follows from Hölder’s inequality that
$$
\begin{equation*}
\begin{aligned} \, M_1^f(r^{1/(2-p)})&=\sum_{k=0}^\infty|a_k|r^{k/p}r^{2k(p-1)/(p(2-p))} \\ &\leqslant\biggl(\,\sum_{k=0}^\infty|a_k|^p r^k\biggr)^{1/p} \biggl(\,\sum_{k=0}^\infty r^{2k/(2-p)}\biggr)^{1-1/p} \\ &=\bigl(M_p^f(r)\bigr)^{1/p}\frac{1}{(1-r^{2/(2-p)})^{(p-1)/p}}\,, \end{aligned}
\end{equation*}
\notag
$$
so that
$$
\begin{equation*}
M_p^f(r) \geqslant \biggl(\frac{1}{\sqrt{1-r^{2/(2-p)}}}-C(\varepsilon) \biggl(\log\frac{1}{1-r^{1/(2-p)}}\biggr)^{3/2+\varepsilon}\biggr)^p (1-r^{2/(2-p)})^{p-1}
\end{equation*}
\notag
$$
for $2^{p/2-1} < r < 1$. Therefore,
$$
\begin{equation*}
M_p^f(r) \geqslant \biggl(\frac{1}{1-r^{2/(2-p)}}\biggr)^{1-p/2}- C_1(\varepsilon)(1-r^{2/(2-p)})^{(p-1)/2} \biggl(\log \frac{1}{1-r^{1/(2-p)}}\biggr)^{3/2+\varepsilon}.
\end{equation*}
\notag
$$
Hence we obtain
$$
\begin{equation*}
M_p(r)-\biggl(\frac{1}{1-r^{2/(2-p)}}\biggr)^{1-p/2} \to 0 \quad\text{as}\ r \to 1
\end{equation*}
\notag
$$
for $1<p<2$. For $p=1$ it is not known whether or not the asymptotic behaviour is similar. This is perhaps a difficult problem. Ricci and Bombieri (see [34]) showed that if we also fix $f(0)=a \geqslant 1/2$, then the corresponding Bohr radius is $1/(2a+1)$. It is easy to show using (10) that if $p \in (1,2)$ and $f(0)=a$, then the generalized Bohr radius of the bounded functions with fixed value at the origin can be calculated by the formula
$$
\begin{equation*}
\frac{1}{r_p(a)}=a^p+\frac{(1-a^2)^p}{1-a^p}, \qquad a \geqslant a_p,
\end{equation*}
\notag
$$
where $a_p$ is the unique root of the equation Note that where $t$ is the unique root of the equation in the interval $(0,1)$. Ali, Barnard, and Solynin [6] considered the following problem: find the Bohr radius in the class of odd functions $f$ such that $|f(z)|\leqslant 1$ for all $z\in \mathbb{D}$. It was shown in [6] that $\sum_{n=0}^{\infty}|a_n|r^{2n+1} \leqslant 1$ for all $|z|=r\leqslant r_*$, where $r_*$ is the unique solution of the equation in the interval $1/\sqrt{3}<r<1$. The value of $r_*$ is approximately $0.7313$.Kayumov and Ponnusami [92] obtained a definitive solution by establishing the following result. Theorem 3.6. The Bohr radius in the class of odd bounded holomorphic functions is precisely $r^*$, where and Note that $r^*\approx 0.789991$. If $f$ and $g$ are analytic functions in $\mathbb{D}$, $\omega$ is a Schwarz function (that is, $\omega$ is analytic in $\mathbb{D}$, $\omega(0)=0$, and $|\omega(z)|<1$ for $|z|<1$), and all three functions satisfy $f(z)=g(\omega(z))$ for $z\in \mathbb{D}$, then we write $f(z)\prec g(z)$ and say that $f$ is subordinate to $g$. For any two analytic functions $f$ and $g$ in $\mathbb{D}$ we say that $f$ is quasisubordinate to $g$ and write $f(z)\prec_{\rm q}g(z)$ if there exists a Schwarz function $\omega$ such that
$$
\begin{equation*}
|f(z)| \leqslant |g(\omega(z))|,\qquad z\in \mathbb{D}.
\end{equation*}
\notag
$$
Bhowmik and Das [32] (the case ‘$\prec$’) and Alkhaleefah, Kayumov, and Ponnusami [7] (the case ‘$\prec_{\rm q}$’) established the following result. Theorem 3.7. Let $f(z)$ and $g(z)$ be analytic functions in $\mathbb{D}$ with Taylor expansions $f(z)=\sum_{k=0}^\infty a_kz^k$ and $g(z)=\sum_{k=0}^\infty b_kz^k$ for $z\in \mathbb{D}$. If $f(z)\prec_{\rm q}g(z)$, then Judging by this result, the following conjecture looks plausible. Conjecture 3.1. Let $f(z)$ and $g(z)$ be analytic functions in $\mathbb{D}$ with Taylor expansions $f(z)=\sum_{k=0}^\infty a_kz^k$ and $g(z)=\sum_{k=0}^\infty b_kz^k$ for $z\in \mathbb{D}$. If $f(z)\prec_{\rm q} g(z)$, then for $p \in [1,2]$ Were this conjecture true, it would link Bohr’s inequality with classical theorems on subordinate analytic functions due to Littelwood and Rogosinski. Again, conformal automorphisms of the disc must play a significant role here: the conjecture is equivalent to automorphisms of this type being extremals in the corresponding extremal problem. As regards Bohr-type inequalities for subordinate functions, a number of interesting results were obtained in [77] and [130]. Now let $S_r(f)$ denote the area of the image of the disc $|z|<r$ under the mapping $f$. The following result [93] was established in 2018. Let $f(z)=\sum_{k=0}^\infty a_k z^k$ be analytic in $\mathbb{D}$, and let $|f(z)| \leqslant 1$ в $\mathbb{D}$. Then
$$
\begin{equation}
\sum_{k=0}^\infty|a_k|r^k+\frac{16}{9}\,\frac{S_r}{\pi} \leqslant 1\quad\text{for } r \leqslant \frac{1}{3}
\end{equation}
\tag{11}
$$
and the quantities $1/3$ and $16/9$ are best possible. The following statement [87] shows that we can add a positive term to the left-hand side of (11). Theorem 3.8. Let $f(z)=\sum_{k=0}^\infty a_k z^k$ be an analytic function in $\mathbb{D}$ such that $|f(z)| \leqslant 1$ in $\mathbb{D}$. Then This is a definitive result in the following sense: there exists an explicit extremal function maximizing the corresponding functional. Expectedly, an automorphism of the disc is this extremal function again. For a holomorphic function $f(z)=\sum_{n=0}^{\infty}a_n z^n$ in $\mathbb{D}$ the Cesàro operator is defined by
$$
\begin{equation*}
{\mathcal C}f(z):=\sum_{n=0}^{\infty}\biggl(\frac{1}{n+1} \sum_{k=0}^{n} a_k \biggr)z^n=\int_{0}^{1}\frac{f(tz)}{1-tz}\, dt.
\end{equation*}
\notag
$$
If $|f|\leqslant 1$, then the Bohr sum can be replaced by
$$
\begin{equation*}
{\mathcal C}_f(r):=\sum_{n=0}^{\infty} \biggl(\frac{1}{n+1}\sum_{k=0}^{n}|a_k|\biggr)r^n.
\end{equation*}
\notag
$$
Note that for each $|z|=r \in [0,1)$ and each holomorphic function $f$ such that $|f(z)|\leqslant 1$ in $\mathbb{D}$ we have the inequality
$$
\begin{equation*}
|{\mathcal C}f(z)| \leqslant \frac{1}{r} \log \frac{1}{1-r}\,.
\end{equation*}
\notag
$$
The theorem below is due to Kayumov, Khammatova, and Ponusamy [91]. It is an analogue of Bohr’s theorem for the Cesàro operator. Theorem 3.9. Let $f$ be a holomorphic function such that $|f|\leqslant 1$ in $\mathbb{D}$, and let $f(z)=\sum_{n=0}^{\infty} a_n z^n$. Then for $r \leqslant R$, where $R=0.5335\ldots$ is the positive root of the equation The value of $R$ is best possible. The question on the behaviour of ${\mathcal C}_f(r)$ as $r\to 1$ arises in a natural way. If $f$ is a holomorphic function such that $|f|\leqslant 1$ in $\mathbb{D}$ and $f(z)=\sum_{n=0}^{\infty} a_n z^n$, then the following inequality holds for $r \in [0,1)$ [91]:
$$
\begin{equation*}
{\mathcal C}_f(r) \leqslant \frac{1}{r}\,\sqrt{\frac{1+r}{1-r}\log(1+r)+\log(1-r)}\,.
\end{equation*}
\notag
$$
Now we state two unsolved problems of certain interest. Here the function $M_1(r)$ is defined by equality (9) for $p=1$. The following result is related to orientation-preserving harmonic maps of the disc. Recall that the class ${\mathcal H}$ of complex-valued harmonic functions $f=h+\overline{g}$ in the unit disc ${\mathbb D}$ and injective subclasses of it are now well understood. Here $h$ and $g$ are holomorphic functions in $\mathbb{D}$ with Taylor expansions
$$
\begin{equation*}
h(z)=\sum_{k=0}^{\infty}a_kz^k\quad\text{and}\quad g(z)=\sum_{k=1}^{\infty}b_kz^k.
\end{equation*}
\notag
$$
The Jacobian determinant of $f$ can easily be calculated: The harmonic map $f$ is said to be orientation preserving if $J_f(z)> 0$ in $\mathbb{D}$. The map $\omega(z)=g'(z)/h'(z)$ is called the complex dilatation of $f=h+\overline{g}$. By Lewy’s classical theorem $f$ is locally univalent and orientation preserving in $\mathbb{D}$ if and only if $|\omega (z)|<1$ for $z\in \mathbb{D}$. The reader can find comprehensive surveys of the geometric and analytic properties of planar harmonic maps in [40] and [63]. Let
$$
\begin{equation*}
f(z)=h(z)+\overline{g(z)}=\sum_{k=0}^\infty a_k z^k+\overline{\sum_{k=1}^\infty b_k z^k}
\end{equation*}
\notag
$$
be an orientation-preserving harmonic map of $\mathbb{D}$ such that the function $h$ is bounded in $ \mathbb{D}$. The theorem below was proved in [94]. Theorem 3.10. If $p \in [0,2]$, then the following inequality is best possible: for all $r \leqslant (2^{1/(p-2)}+1)^{p/2-1}$. For $p>2$ the following inequality holds: In particular, the following inequalities are straightforward consequences for $p= 1$ [95]:
$$
\begin{equation*}
|a_0|+\sum_{k=1}^\infty (|a_k|+|b_k|) r^k \leqslant \frac{\|h\|_{\infty}}{r}\,(5-2\sqrt{6}\,\sqrt{1-r^2}\,) \quad\text{for}\ \frac{1}{5} \leqslant r \leqslant \sqrt{\frac{2}{3}}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
|a_0|+\sum_{k=1}^\infty(|a_k|+|b_k|)r^k \leqslant \|h\|_{\infty} \quad\text{for}\ r \leqslant \frac{1}{5}\,.
\end{equation*}
\notag
$$
A number of interesting inequalities for various classes of harmonic maps of the disc can be found in [84] and [2].
4. Hardy-type inequalities in domainsIn this section we turn back to estimates for integral means in plane domains, but these means will now be of slightly other nature, although they will still be closely related to the geometry of plane domains. We look at arbitrary plane domains $\Omega \subset \mathbb{C}$, $\Omega \ne \mathbb{C}$. The distance to the boundary of the domain
$$
\begin{equation*}
\rho(z,\partial\Omega):=\inf_{w\in \mathbb{C}\setminus \Omega}|z-w|,\qquad z\in \Omega,
\end{equation*}
\notag
$$
plays a central role in Hardy-type inequalities (note that in the previous section we described results whose proofs were also based on the analytic and geometric properties of this function). Note that the distance function is widely used in the considerations of various problems in complex analysis and mathematical physics. This function has a simple geometric meaning and a number of remarkable properties. One of these properties is that in each domain $\Omega \subset {\mathbb{C}}$, $\Omega \ne {\mathbb{C}}$, the distance function is Lipschitz continuous:
$$
\begin{equation*}
|\rho(z,\partial\Omega)-\rho(\zeta,\partial \Omega)| \leqslant |z-\zeta| \quad \forall\,z, \zeta \in \Omega.
\end{equation*}
\notag
$$
Yet another important property of the distance function, which is often used, is related to conformal isomorphisms of the disc: if $f$ is a univalent conformal mapping of the disc $\mathbb{D}$ onto a simply connected domain $\Omega$, then we have the following inequalities:
$$
\begin{equation*}
\frac{1}{4} \,|f'(z)|(1-|z|^2) \leqslant \rho(f(z),\partial\Omega) \leqslant |f'(z)|(1-|z|^2), \qquad z \in \mathbb{D}.
\end{equation*}
\notag
$$
The left-hand inequality follows from Koebe’s $1/4$-theorem, while the right-hand one follows from Schwarz’s lemma (for instance, see [74], Chap. 2, § 4, and Chap. 8, § 1). Recall that $|f'(z)|(1-|z|^2)= R(w,\Omega)$ is the conformal radius of the domain $\Omega$ at the point $w=f(z) \in \Omega$. Given a function $u\colon \Omega \to \mathbb{R}$, we use the notation $u=u(z)$ and the formula
$$
\begin{equation*}
\nabla u(z)=\frac{\partial u(z)}{\partial x}+ i \,\frac{\partial u(z)}{\partial y}\,, \qquad z=x+iy\in \Omega.
\end{equation*}
\notag
$$
Consider the Hardy-type inequality
$$
\begin{equation}
\iint_\Omega\frac{|\nabla u(z)|^p}{\rho^{s-p}(z,\partial\Omega)}\,dx\,dy \geqslant c_p(s,\Omega)\iint_\Omega\frac{|u(z)|^p} {\rho^s(z,\partial\Omega) }\,dx\,dy \quad \forall\,u\in C_0^1(\Omega),
\end{equation}
\tag{13}
$$
where the numbers $p\in [1,\infty)$ and $s \in \mathbb{R}$ are fixed. Here $C_0^1(\Omega)$ is the space of continuously differentiable functions with compact support in $\Omega$. We assume that the constant $c_p(s,\Omega)\in [0,\infty)$ is sharp, that is, it is the greatest possible constant in this inequality. It should be noted that this optimal constant $c_p(s,\Omega)$ in (13) is dimensionless and invariant under linear maps of the domain: $c_p(s,\Omega)=c_p(s,a\Omega+b)$, where $a \Omega+b=\{a z+b\colon z \in \Omega\}$, $a \ne 0$. The main problem is as follows: describe all domains such that $c_p(s,\Omega)>0$ in reasonable terms (for example, in terms of Euclidean geometry). This problem has proved to be quite a non-trivial one, and so far it has been solved in the following cases (see [26] and [18], and also [10] and [19]): 1) if $s>2$, then $c_p(s, \Omega)>0$ for each domain $\Omega \subset {\mathbb{C}}$, $\Omega \ne \mathbb{C}$; 2) if $s=2$, then $c_p(2,\Omega)>0$ if and only if the domain $\Omega \subset \mathbb{C}$, $\Omega \ne \mathbb{C}$, has a uniformly perfect boundary. Recall [31] that a compact subset $F$ of $ \overline{\mathbb{C}}$ is said to be uniformly perfect if $F$ contains at least two points and the moduli of conformal annuli lying in $\overline{\mathbb{C}}\setminus F$ and separating $F$ are uniformly bounded. If the domain is simply connected, then its boundary is uniformly perfect by definition. In some cases it is not enough to know that the quantity $c_p(s,\Omega)$ is positive. One would also like to have explicit bounds for this quantity in terms of geometric characteristics of the domain. Here we can point out the following beautiful result, which several authors proved independently of one another (see [49], [10], [106], and [83]): it turns out that $c_2(2,\Omega)=1/4$ for each convex domain $\Omega \subset {\mathbb{C}}$, $\Omega \ne {\mathbb{C}}$. It is natural to conjecture that the Hardy constant for an arbitrary domain does not exceed the Hardy constant for convex domains, such as a disc or a half-plane. A topical famous problem (Davies’s conjecture [49], [50]) is as follows: show that for an arbitrary domain $\Omega \subset {\mathbb{C}}$, $\Omega \ne {\mathbb{C}}$, Inequality (14) was proved by Davies for domains $\Omega \subset {\mathbb{C}}$ with at least one boundary point $y_0\in \partial \Omega$ that is ‘regular’ in a certain sense. Namely, $y_0$ is regular if there exists a neighbourhood $U(y_0)$ such that the intersection $U(y_0)\cap \partial \Omega$ is a smooth arc. Some weaker versions of this condition of ‘regularity’ for a point $y_0\in \partial \Omega$ are possible, but one knows of no proof of (14) without some or other additional conditions. In fact, all known methods for constructing a minimizing sequence $u_k\in C_0^1(\Omega)$ ($k=1,2,\dots$) are based on a certain prescribed behaviour of the distance function $\rho(z,\partial\Omega)$ on some set $U(y_0)\cap \Omega$. Also note that the above problem is still unsolved even for the system of all simply connected domains $\Omega \subset {\mathbb{C}}$ that are conformally equivalent to the unit disc. At this point we can only say that for each domain $\Omega \subset {\mathbb{C}}$, $\Omega \ne {\mathbb{C}}$, we have $c_2(2, \Omega)\leqslant 1$ (this is a consequence of conformally invariant inequalities from the spectral theory of Beltrami’s equation, a remarkable formula due to Elstrodt, Patterson, and Sullivan and the principle of the hyperbolic metric: see [18], p. 102). The following challenging task is related to this problem: describe geometrically the set of domains $\Omega \subset {\mathbb{C}}$, $\Omega \ne {\mathbb{C}}$, such that We know a number of non-convex domains for which (15) holds. Here are two examples.Davies’s example [49], [50]: the constant $c_2(2,\Omega_{\beta})$ is equal to $1/4$ for the sector
$$
\begin{equation*}
\Omega_{\beta}=\{z=r e^{i\theta} \in {\mathbb C}\colon 0< r < 1, 0<\theta<\beta\}
\end{equation*}
\notag
$$
if and only if $\beta \leqslant \beta_0\approx 4.856$. Davies found the critical vale of the angle numerically, but an explicit formula can also be indicated (for instance, see [10]):
$$
\begin{equation*}
\beta_0=3\pi-4 \arctan \frac{\Gamma^4(1/4)}{8\pi^2}\,.
\end{equation*}
\notag
$$
Avkhadiev established the following result [13]: for an annulus the constant $c_2(2,A_{rR})$ is equal to $1/4$ if and only if $R/r \leqslant c^*\approx 36.6$. The critical value $c^*\approx 36.6$ of the ratio of radii can be determined from a certain transcendental equation involving Gauss hypergeometric functions.In addition, a rich family $\Theta_{1/4}(2)$ of non-convex domains $\Omega \subset {\mathbb{C}}$, $\Omega \ne {\mathbb{C}}$, with property (15) was described in [10]. It contains domains of any connectivity and has a direct relation to domains satisfying the exterior sphere condition. Note also [19], where 15 difficult unsolved problems relating to inequalities of Hardy–Rellich and Poincaré–Friederichs type were presented. Now we describe Maz’ya’s, Ancona’s, and Fernández’s criteria for the constants in the basic Hardy-type inequalities to be positive. Ancona’s and Fernández’s criteria relate to two-dimensional domains. First we look at Maz’ya’s criterion, which treats domains of dimension $n\geqslant 2$. Let $p\in (1,\infty)$. In a domain $\Omega \subset {\mathbb{R}^n}$, $\Omega \ne {\mathbb{R}^n}$, for functions $u\colon\Omega \to \mathbb{R}$ consider the following Hardy-type inequality:
$$
\begin{equation}
\int_{\Omega}{|\nabla u(x)|^p\,dx}\geqslant c_p(\Omega)\int_{\Omega} \frac{|u(x)|^p\,dx}{\rho^{p}(x,\partial\Omega)} \quad \forall\,u \in C_0^{1} (\Omega),
\end{equation}
\tag{16}
$$
where $\rho(x,\partial\Omega)$ is the distance from $x=(x_1,x_2,\dots,x_n) \in \Omega$ to the boundary of the domain, $d x=dx_1\,dx_2 \cdots dx_n$ is a measure element, and the constant $c_p(\Omega)\in [0, \infty)$ is by definition the greatest constant possible here. In the case of inequality (16) we can state Maz’ya’s criterion as follows. Let $\operatorname{Cap}_p (\mathbb{K}, \Omega)$ be the $p$-capacity of the compact subset $\mathbb{K}$ of $\Omega$, which is defined by
$$
\begin{equation*}
\operatorname{Cap}_p(\mathbb{K},\Omega)=\inf\biggl\{\int_{\Omega} |\nabla u(x)|^p\,dx\colon u \in C_0^{\infty} (\Omega), \ u(x)\geqslant 1 \ \forall\,x \in \mathbb{K}\biggr\}.
\end{equation*}
\notag
$$
Theorem 4.1 ([107], 1985). Let $p\in (1, \infty)$. Then $c_p(\Omega)$ is positive if and only if there exists a finite constant $S_p=S_p(\Omega)$ such that for each compact set $\mathbb{K}\subset \Omega$ For $p>n$ we can write out explicit estimates for the constant in Maz’ya’s isoperimetric inequality in an arbitrary domain. To do this we use the following result due to Avkhadiev. Theorem 4.2 ([9], 2006). Let $p\in [1,\infty)$, $n\geqslant 2$, $s\in [n,\infty)$, and let $\Omega \subset {\mathbb{R}^n}$ be an arbitrary domain satisfying the single condition $\Omega \ne {\mathbb{R}^n}$. Then the following Hardy-type inequality holds: There exist domains $\Omega\subset\mathbb{R}^n$, $\Omega\ne\mathbb{R}^n$, for which $(s-n)^p/p^p$ is the greatest possible constant in this inequality. Theorems 4.1 and 4.2 yield the following result in an obvious way. Corollary 4.2.1. Let $n\geqslant 2$, and let $\Omega \subset {\mathbb{R}^n}$ be a domain such that $\Omega \ne {\mathbb{R}^n}$. If $p\in (n,\infty)$, then the following Maz’ya isoperimetric inequality holds for each compact set $\mathbb{K}\subset \Omega$: Clearly, the value of $S_p(\Omega)$ and Corollary 4.2.1 can be refined for a number of domains for which we know the sharp values of $c_p(\Omega)$. Here is an example. It is known that for each $p\in (1,\infty)$ we have $c_p(\Omega)={(p-1)^p}/{p^p}$ for all convex domains $\Omega \subset {\mathbb{R}^n}$, $\Omega \ne {\mathbb{R}^n}$ (see [26] and [19]). Hence the following is true. Corollary 4.2.2. For $n\geqslant 2$ let $\Omega \subset {\mathbb{R}^n}$ be a convex domain such that $\Omega \ne {\mathbb{R}^n}$. Then for each $p\in (1,\infty)$ and each compact set $\mathbb{K}\subset \Omega$ Note that Miklyukov and Vuorinen [111] (1999) considered generalized Hardy- type inequalities on Riemannian manifolds of dimension $n\geqslant 2$ and proved an analogue of Theorem 4.1 for these inequalities. In place of $p$-capacity, they used certain geometric quantities related to the isoperimetric profile of the Riemannian manifold. In what follows we use the complex variables $z,w \in \mathbb{C}$. In addition to Maz’ya’s criterion, we use Ancona’s and Fernandez’s criteria for the positivity of Hardy constants. These two criteria are only related to two-dimensional domains, and only the case $p=n=2$ in (16) is considered, that is, in $\Omega \subset {\mathbb{C}}$ we look at the inequality
$$
\begin{equation}
\iint_{\Omega}|\nabla u(z)|^2\,dx\,dy\geqslant c_2(\Omega)\iint_{\Omega} \frac{|u(z)|^2\,dx\,dy}{\rho^2(z,\partial\Omega)} \quad \forall\,u \in C_0^1(\Omega),
\end{equation}
\tag{18}
$$
where $\rho(z,\partial\Omega)$ is the distance of the point $z=x+iy \in \Omega$ to the boundary of the domain and the constant $c_2(\Omega)\in [0,\infty)$ is defined as the greatest possible constant here. We set
$$
\begin{equation*}
B_z(r)=\{w\in \mathbb{C}\colon |w-z|<r\}\quad\text{and} \quad \partial B_z(r)=\{w\in \mathbb{C}\colon |w-z|=r\}
\end{equation*}
\notag
$$
and present a definition due to Ancona [8], which we need below. Definition 4.1. Let $\Omega \subset {\mathbb{C}}$ be a domain such that $\Omega \ne {\mathbb{C}}$. For each point $z \in (\partial \Omega)\setminus \{\infty\}$ and each radius $r>0$, consider the harmonic measure of $\Omega\cap \partial B_z(r)$ in the open set $\Omega\cap B_z(r)$. We say that the boundary of $\Omega$ is uniformly $\Delta$-regular if there exists a constant $\varepsilon_1(\Omega) \in (0,1)$ such that for all $z \in (\partial\Omega)\setminus \{\infty\}$ and $r>0$ this harmonic measure $\omega_{zr}(w)$, $w\in \Omega\cap B_z(r)$, satisfies the inequality for all $w\in \Omega \cap \partial B_z(r/2)$.Recall that harmonic measures are the subject of the monograph [73] by Garnett and Marshall. Ancona’s criterion is formulated as follows. Theorem 4.3 ([8], 1986). Let $\Omega \subset {\mathbb{C}}$ be a domain such that $\Omega \ne {\mathbb{C}}$. Then $c_2(\Omega)>0$ if and only if the boundary of $\Omega$ is uniformly $\Delta$-regular. The following lemma of Ancona is essential for the proof. Lemma 4.1 ([8], 1986). Let $\Omega \subset {\mathbb{C}}$ be a domain such that $\Omega \ne {\mathbb{C}}$. Then $c_2(\Omega)>0$ if and only if there exist a positive superharmonic function $v(z)$ in $\Omega$ and a positive constant $\varepsilon=\varepsilon(\Omega)$ such that in $\Omega$ in the distributional sense, that is, for any non-negative function $\psi \in C_0^{\infty}(\Omega)$ The greatest possible value of $\varepsilon=\varepsilon(\Omega)$ is equal to $c_2(\Omega)$. In its turn, the verification of this lemma is based to a substantial extent on the Lax–Milgram theorem on the representation of a non-negative, continuous, coercive bilinear form in a Hilbert space. Let $\Omega\subset\mathbb{C}$ be a domain of hyperbolic type, that is, with at least three boundary points. Then at each point in the domain we know the coefficient $\lambda_\Omega(z)$ of the Poincaré metric with Gaussian curvature $k=-4$, and we know the hyperbolic (conformal) radius $R(z,\Omega)=1/\lambda_\Omega(z)$. It is well known that the hyperbolic radius satisfies Liouville’s equation $R(z,\Omega)\Delta R(z,\Omega)=|\nabla R(z,\Omega)|^2-4$ and, at each point $z\in \Omega$, we have the inequality which is a simple consequence of the hyperbolic metric principle.Beardon and Pommerenke considered the quantity
$$
\begin{equation*}
\alpha(\Omega):=\inf_{z\in \Omega} \frac{\rho(z,\partial\Omega)}{R(z,\Omega)}\,.
\end{equation*}
\notag
$$
They proved the following theorem. Theorem 4.4 ([31], 1978). For a domain $\Omega\subset\mathbb{C}$ of hyperbolic type the quantity $\alpha(\Omega)$ is positive if and only if each doubly connected domain $G \subset \Omega$ separating the boundary of $\Omega$ has conformal modulus at most $L=L(\Omega)$. In what follows $M(\Omega)$ is the smallest of the constants $L= L(\Omega)$. We call $M(\Omega)$ the conformal maximum modulus. For each simply connected domain $\Omega\subset\mathbb{C}$ of hyperbolic type we have $\alpha(\Omega)>0$ and set $M(\Omega)=0$ (also see Definition 5.1 below). With this convention there is no need to distinguish the case of simply connected domains in the statements of some results. Fernández’s criterion below was established with the help of the Beardon–Pommerenke theorem also Ancona’s lemma stated below. Theorem 4.5 ([69], 1989). Let $\Omega \subset {\mathbb{C}}$ be a domain of hyperbolic type. Then $c_2(\Omega)$ is positive if and only if $M(\Omega)< \infty$. In particular, Ancona [8] proved that for each simply connected domain $\Omega \subset{\mathbb{C}}$ of hyperbolic type we have $c_2(\Omega)\geqslant 1/16$. In the next section we present the definition of the Euclidean maximum modulus $M_0(\Omega)$, which is easier to evaluate or estimate than the quantity $M(\Omega)$. 5. Conformal and Euclidean maximum moduliRecall that, for a doubly connected domain $\Omega_2\subset\overline{\mathbb{C}}$ which is conformally equivalent to an annulus
$$
\begin{equation*}
A(\Omega_2)=\{z \in {\mathbb{C}} \colon a<|z|< b\}, \qquad 0\leqslant a<b \leqslant \infty,
\end{equation*}
\notag
$$
the conformal modulus is defined by
$$
\begin{equation*}
\operatorname{Mod}(\Omega_2)=\frac{1}{2\pi}\log\frac{b}{a} \in (0,\infty]
\end{equation*}
\notag
$$
under the natural convention that $\operatorname{Mod}(\Omega_2)=\infty$ for $a=0$ or $b=\infty$. Now we give the general definition of the conformal maximum modulus $M (\Omega)$ of the domain $\Omega \subset \overline{\mathbb{C}}$. Definition 5.1. Let $\Omega \subset \overline {\mathbb{C}}$ be a domain with boundary containing at least two points. Then the conformal maximum modulus $M(\Omega)$ is defined as follows. 1) If $\Omega$ is a simply connected domain, then set $M(\Omega)=0$. 2) If $\Omega$ is a doubly connected domain, then $M(\Omega)$ is its conformal modulus, that is,
$$
\begin{equation*}
M(\Omega):=\operatorname{Mod}(\Omega)= \frac{1}{2\pi}\log\frac{b}{a} \in (0,\infty],
\end{equation*}
\notag
$$
provided that $\Omega$ is equivalent to the annulus $\{z \in\mathbb{C}\colon a<|z|< b\}$, $0\leqslant a<b\leqslant\infty$. 3) If $\Omega$ is multiply connected, then where the upper bound is taken over all doubly connected domains $\Omega_2$ such that $\Omega_2 \subset \Omega$ and $\Omega_2$ separates the boundary of $\Omega$, so that the set $\overline{\Omega} \setminus \Omega_2$ is disconnected.For a doubly connected domain we can also define $M(\Omega)$ using part 3) of the definition. This is not in contradiction with part 2), because it is well known that the modulus of a doubly connected domain is non-decreasing under extensions. We comment on part 1) of the definition. Assume that the boundary of the domain $\Omega \subset \overline {\mathbb{C}}$ contains at least two points and the domain is simply connected. Hence it is conformally equivalent to a disc. In this case we set $M(\Omega)=0$ by definition. This does not follow from any facts, but is a convenient convention to avoid distinguishing the case of simply connected domains in a number of statements (cf. the convention $0!=1$). To define the Euclidean maximum modulus $M_0 (\Omega)$ we need the set $\mathbb{A}\mathrm{nn}(\Omega)$ of annuli
$$
\begin{equation*}
A= A(z_0;a,\,b):=\{z\in {\mathbb{C}}\colon a<|z-z_0| < b\},
\end{equation*}
\notag
$$
with the following properties: (i) $A(z_0; a,\, b)$ lies in $\Omega$, where $0<a<b<\infty$; (ii) the centre $z_0$ of the annulus lies on the boundary $\partial\Omega$; (iii) $A(z_0; a,\, b)$ separates the boundary of $\Omega$, so that each of the two domains
$$
\begin{equation*}
\{z\in \mathbb{C}\colon |z-z_0|<a\}, \quad \{z\in \overline{\mathbb{C}}\colon |z-z_0|> b\}
\end{equation*}
\notag
$$
contains at least one boundary point of $\Omega$. Of course, the set $\mathbb{A}\mathrm{nn}(\Omega)$ can also be empty. Definition 5.2. Let $\Omega \subset \overline {\mathbb{C}}$ be a domain with at least two boundary points. Let $\mathbb{A}\mathrm{nn}(\Omega)$ be the set of annuli introduced above. Then the Euclidean maximum modulus ь $M_0(\Omega)$ is defined as follows. 1) If $\mathbb{A}\mathrm{nn}(\Omega)=\varnothing$, then $M_0(\Omega)=0$. 2) If $\mathbb{A}\mathrm{nn}(\Omega)$ is not empty, then If $M_0(\Omega)<\infty$, then, following Pommerenke [128], we say that the boundary of $\Omega$ is a uniformly perfect set. By definition the conformal modulus $M(\Omega)$ vanishes if and only if $\Omega$ is a simply connected domain conformally equivalent to the unit disc. The definition of $M_0(\Omega)$ is not related to conformal mappings. The following example from [19] shows that $M_0 (\Omega)$ can also vanish for multiply connected domains. Example 5.1. Let $\mathbb{K}$ be the classical Cantor set on the interval $[0,1]$, and let $\Omega_0:=\{x+iy\in \mathbb{C}\colon|x|<\infty,|y|<1\}$. Consider the multiply connected domain obtained by removing a ‘fence’ of line segments ‘nailed’ to the points in the classical Cantor set from the strip $\Omega_0$. Then because $\mathbb{A}\mathrm{nn}(\Omega(\mathbb{K}))= \varnothing$. It follows from the definitions that $0\leqslant M_0(\Omega)\leqslant M(\Omega)$. Carleson and Gamelin [38] (1993) established the following remarkable property:
$$
\begin{equation*}
M_0(\Omega)<\infty\ \ \Longleftrightarrow\ \ M(\Omega)<\infty.
\end{equation*}
\notag
$$
Note that this equivalence was refined by several authors. In particular, it was proved by Avkhadiev and Wirths in [25] (2009) that
$$
\begin{equation*}
M_0(\Omega)\leqslant M(\Omega)\leqslant M_0(\Omega)+\frac{1}{2}
\end{equation*}
\notag
$$
for any domain $\Omega \subset {\mathbb{C}}$ with at leat two boundary points. The following analogue of this inequality holds for domains $\Omega \ni \infty$ (see [11], 2015):
$$
\begin{equation*}
M_0(\Omega)\leqslant M(\Omega)\leqslant 2 M_0(\Omega)+1
\end{equation*}
\notag
$$
for each domain $\Omega \subset \overline{\mathbb{C}}$ with at least two points on the boundary. For domains $\Omega \subset \mathbb{C}$ (and only for such domains) we have the following fact, going back to Beardon and Pommerenke [31]. Let $\Omega\subset {\mathbb{C}}$ be a domain of hyperbolic type. Then the following inequality holds:
$$
\begin{equation}
\max_{z\in \Omega}\frac{R(z,\Omega)}{\rho(z,\partial\Omega)} \leqslant 2\pi M_0(\Omega)+\frac{\Gamma^4(1/4)}{2\pi^2}\,.
\end{equation}
\tag{20}
$$
It can be found by [25] and refines estimates from the paper [31] mentioned above. Here is a further result on the relationship between the coefficient $\lambda_\Omega(z)$ of the Poincaré metric and the hyperbolic radius $R(z,\Omega):= 1/\lambda_\Omega(z)$ with conformal mappings from the unit disc onto a domain $\Omega$. Theorem 5.1 ([17], 2019). Let $\Omega\subset \overline{\mathbb{C}}$ be a domain of hyperbolic type, and let $f\colon\mathbb{D} \to \Omega$ be a locally conformal covering from the disc $\mathbb{D}=\{z\in \mathbb{C}\colon |z|<1\}$ onto $\Omega$ (in particular, if $\Omega$ is simply connected, then $f\colon\mathbb{D} \to \Omega$ is a univalent conformal mapping of $\mathbb{D}$ onto $\Omega$). Then where $z=f(\zeta) \in \Omega$ and $S_f(\zeta)$ is the Schwarzian derivative of $f$ at $\zeta \in \mathbb{D}$. Calculating the precise value of $\operatorname{Mod}(\Omega)$ is usually a complicated task. We present an example based on well-known results of Teichmüller and Ahlfors (see [4]). Example 5.2. Let $t\in (0,\infty)$ and $H_+=\{z\in \mathbb{C}\colon \operatorname{Re}z>0\}$. Consider the doubly connected domain $\Omega_t= H_+\setminus \bigl[\sqrt{t}\,,\sqrt{t+1}\,\bigr]$. The function $f(z)=t-z^2$ maps $\Omega_t$ univalently and conformally onto the ‘Teichmüller ring’ $A_t:=\mathbb{C}\setminus([-1,0]\cup [t,\infty))$. Hence $\operatorname{Mod}(\Omega_t)=\operatorname{Mod}(A_t)$, $\operatorname{Mod}(\Omega_1)=1/2$ and the following formula, due to Ahlfors (see [4]), holds: It is known that $\operatorname{Mod}(\Omega_t)=\operatorname{Mod}(A_t)$ is a monotonically increasing function of $t\in (0,\infty)$, and we have Let $\varepsilon \in (0,\infty)$, $t \in (0,\infty)$, and let $\varepsilon\Omega_t= H_+\setminus \bigl[\varepsilon\sqrt{t}\,, \varepsilon\sqrt{t+1}\,\bigr]$ be the image of $\Omega_t$ under the conformal mapping $w=\varepsilon z$. Then $\operatorname{Mod}(\Omega_t)>0$ and we have where6. The inequality $M_0(\Omega)<\infty$ is a positivity criterion for a number of constantsIn this section we describe results due to Avkhadiev and connected with a criterion for the positivity of the constants in a number of Hardy–Rellich type inequalities and some related estimates. The first result is related to a parametric inequality which generalizes (18). Let $p\in [1,\infty)$. In a domain $\Omega \subset {\mathbb{C}}$, $\Omega \ne {\mathbb{C}}$, we look at the following Hardy-type inequality for functions $u\colon\Omega \to \mathbb{R}$:
$$
\begin{equation}
\iint_{\Omega} \frac{|\nabla u(z)|^p\,dx\,dy}{\rho^{2-p}(z,\partial\Omega)} \geqslant c_p(2, \Omega)\iint_{\Omega}\frac{|u(z)|^p\,dx\,dy} {\rho^2(z,\partial\Omega)} \quad \forall\,u \in C_0^1(\Omega),
\end{equation}
\tag{22}
$$
where $c_p(2,\Omega)\in [0,\infty)$ is by definition the greatest constant possible here. It is clear that $c_2(2,\Omega)=c_2(\Omega)$, where $c_2(\Omega)$ is the greatest possible constant in (16). Hence Fernández’s criterion is equivalent to the following assertion. The constant $c_2(2,\Omega)$ is positive if and only if the boundary of $\Omega$ is a uniformly perfect set. Avkhadiev discovered a new proof for this result and also generalized it to each $p\in [1,\infty)$. In particular, this proof does not use Ancona’s lemma mentioned above. The following two-sided estimate can be established:
$$
\begin{equation*}
\frac{1}{16(\pi M_0(\Omega)+\gamma_0)^{4}}\leqslant c_2(2,\Omega)=c_2(\Omega)\leqslant \frac{1}{4 M_0^2(\Omega)}\quad \biggl(\gamma_0=\frac{\Gamma^{4}(1/4)}{4\pi^2}\biggr).
\end{equation*}
\notag
$$
These estimates show that the constant $c_2(2,\Omega)=c_2(\Omega)$ is positive if and only if $M_0(\Omega)<\infty$. To state the criterion we need both bounds, but the lower one is particularly interesting because the minorant participates in an inequality. Namely, the following result holds. Proposition 6.1. Let $\Omega\subset\mathbb{C}$, $\Omega\ne\mathbb{C}$, be a domain. If the Euclidean maximum modulus satisfies $M_0(\Omega)<\infty$, then For the domain $\Omega(\mathbb{K})$ in Example 5.1, which is defined in (19), the Euclidean maximum modulus $M_0(\Omega(\mathbb{K}))$ is zero. As a consequence, in this domain
$$
\begin{equation*}
\iint_{\Omega(\mathbb{K})}|\nabla u(z)|^2\,dx\,dy\geqslant \frac{16\pi^8}{\Gamma^{16}(1/4)}\iint_{\Omega(\mathbb{K})} \frac{|u(z)|^2\,dx\,dy}{\rho^2(z,\partial\Omega(\mathbb{K}))} \quad \forall\,u \in C_0^{1}(\Omega(\mathbb{K})).
\end{equation*}
\notag
$$
Now we state the main result concerning (22). Theorem 6.1 ([9], 2006). Let $p\in [1,\infty)$, and let $\Omega\subset\mathbb{C}$ be a domain such that $\Omega\ne\mathbb{C}$. Then so that $c_p(2,\Omega)$ is positive if and only if the boundary of $\Omega\subset\mathbb{C}$ is a uniformly perfect set. Then the following inequality holds: Now consider the following Rellich-type inequality:
$$
\begin{equation}
\iint_\Omega\frac{|\Delta u(z)|^2\,dx\,dy}{\rho^{s-4}(z,\partial\Omega)} \geqslant C^*_2(s,\Omega)\iint_\Omega\frac{|u(z)|^2\,dx\,dy} {\rho^s(z,\partial\Omega)} \quad \forall\,u\in C_0^\infty(\Omega),
\end{equation}
\tag{23}
$$
where $s \in \mathbb{R}$ is a fixed number, and $C^*_2(s,\Omega)\in [0,\infty)$ is by definition the greatest possible constant in (23). Theorem 6.2 ([12], 2016). Let $\Omega \subset {\mathbb{C}}$ be a domain such that $\Omega \ne {\mathbb{C}}$. Then Open problem. Prove or disprove the following assertion: if $m\in \mathbb{Z}\setminus\{1,2\}$, then
$$
\begin{equation*}
M_0(\Omega)<\infty \ \ \Longleftrightarrow \ \ C^*_2(2m,\Omega)>0
\end{equation*}
\notag
$$
on the set of domains $\Omega \subset {\mathbb{C}}$ such that $\Omega \ne {\mathbb{C}}$. The following implication was proved in [16]:
$$
\begin{equation*}
M_0(\Omega)<\infty \ \ \Longleftarrow\ \ C^*_2(2m,\Omega)>0.
\end{equation*}
\notag
$$
Let $m\geqslant 2$ be a fixed integer, and let $\Delta^{m/2}$ be the polyharmonic operator defined by
$$
\begin{equation*}
\Delta^{m/2}u:=\begin{cases} \Delta^j u &\text{if $m=2j$ is even}, \\ \nabla(\Delta^j u) &\text{if $m=2j+1$ is odd}, \end{cases}
\end{equation*}
\notag
$$
where $u$ is a smooth real function in some domain. Consider the Rellich-type inequality
$$
\begin{equation}
\iint_{\Omega}|\Delta^{m/2} u(z)|^2\,dx\,dy\geqslant A_m(\Omega)\iint_{\Omega}\frac{|u(z)|^2\,dx\,dy} {\rho^{2m}(z,\partial\Omega)}\quad \forall\,u\in C_0^\infty(\Omega),
\end{equation}
\tag{24}
$$
where $A_m(\Omega) \in [0,\infty)$ is the greatest constant possible here. It is obvious that $A_2(\Omega)=C^*_2(4,\Omega)$, so Theorem 6.2 is connected directly with the following result. Theorem 6.3 ([14], 2018). Let $m \in \mathbb{N}$, $m\geqslant 2$. For a domain $\Omega \subset \mathbb{C}$, $\Omega \ne \mathbb{C}$, the constant $A_m(\Omega)$ is positive if and only if $\partial\Omega$ is a uniformly perfect set. In addition, Using the estimates from Theorems 6.1 and 6.3 we can prove the following. Proposition 6.2. Let $m \in \mathbb{N}$, $m\geqslant 2$, and let $\Omega\subset\mathbb{C}$ be a domain such that $\Omega\ne\mathbb{C}$. If $M_0(\Omega)<\infty$, then where $\gamma_0={\Gamma^{4}(1/4)}/{(4\pi^2})$. It looks like a promising idea to consider $L_p$-versions of (23) and (24) for $p\in [1,\infty)$ and establish analogues of Theorems 6.2 and 6.3 for the new constants (provided that $p\ne 2$). This could only be partially implemented because the methods used in the proofs of the above theorem do not work for $p\ne 2$. Nevertheless, recently Avkhadiev [20] (2022) obtained analogues of Theorem 6.2 related to $L_p$-versions of inequality (23). Namely, in a domain $\Omega \subset {\mathbb{C}}$, $\Omega \ne {\mathbb{C}}$, consider the following analogues of the Rellich-type inequality (23) for functions $u\colon\Omega \to \mathbb{R}$:
$$
\begin{equation}
\iint_{\Omega} \frac{|\Delta u|^p\,dx\,dy}{\rho^{2-2p}(z,\partial\Omega)} \geqslant C^*_p(2,\Omega)\iint_{\Omega}\frac{|u|^p\,dx\,dy} {\rho^2(z,\partial\Omega)} \quad \forall\,u \in C_0^\infty(\Omega)
\end{equation}
\tag{25}
$$
and
$$
\begin{equation}
\iint_{\Omega}\frac{|\Delta u|^p\,dx\,dy}{\rho^{2-2p}(z,\partial\Omega)} \geqslant C^{**}_p(2,\Omega)\iint_{\Omega}|u|^{p-2}|\nabla u|^2\,dx\,dy \quad \forall\,u \in C_0^\infty(\Omega),
\end{equation}
\tag{26}
$$
where $u=u(z)$ and the quantities $C^{*}_p(2,\Omega)$ and $C^{**}_p(2,\Omega)$ are by definition the greatest possible constants. The following theorem was established. Theorem 6.4 ([20], 2022). Let $p\in [2,\infty)$ and let $\Omega\subset\mathbb{C}$ be a domain such that $\Omega\ne\mathbb{C}$. Then Using inequalities (20), (25), (26), (27), and (28) we obtain the following two results. Proposition 6.3. Let $\Omega\subset\mathbb{C}$ be a domain such that $\Omega\ne\mathbb{C}$. If $p\in [2,\infty)$ and the Euclidean maximum modulus of $\Omega$ satisfies $M_0(\Omega)< \infty$, then for each function $u \in C_0^\infty(\Omega)$. Proposition 6.4. Let $\Omega\subset\mathbb{C}$, $\Omega\ne\mathbb{C}$, be a domain. If $p\in [2,\infty)$ and the Euclidean maximum modulus of $\Omega$ satisfies $M_0(\Omega)< \infty$, then for each function $u \in C_0^\infty (\Omega)$. Next we sketch the proof of a number of refined estimates for the constants in Hardy and Rellich-type inequalities for simply and doubly connected domains. To do this we need several results from geometric function theory. In a domain $\Omega \subset\overline{\mathbb{C}}$ of hyperbolic type we consider the inequality
$$
\begin{equation}
\iint_{\Omega}|\nabla u (z)|^2 \,dx\,dy \geqslant h_2(\Omega) \iint_\Omega\frac{|u(z)|^2\,dx\,dy}{R^2(z,\Omega)} \quad \forall\,u \in C_0^1(\Omega),
\end{equation}
\tag{29}
$$
where $z=x+iy$ and the constant $h_2(\Omega)$ is sharp, that is, the greatest possible one. This inequality is a conformally invariant analogue of the Poincaré–Friederichs inequality and is used in hyperbolic geometry (for instance, see Sullivan [144]). From the variety of results related to (29), now we only need the following fact (for instance, see Fernández [69] or Fernández and Rodrigues [70]): for any simply or doubly connected domain $\Omega \subset\overline{\mathbb{C}}$ of hyperbolic type the equality $h_2(\Omega)=1$ holds. We present a generalization of this result. For each $p\in [1,\infty)$, in simply and doubly connected domains we have a conformally invariant inequality generalizing (29), which can be stated as follows. Theorem 6.5 (see [11]). If $p\in [1,\infty)$ and $\Omega$ is a simply or doubly connected domain of hyperbolic type, then for each real-valued function $u\in C_0^\infty(\Omega)$ where the constant $2^p/p^p$ is sharp, that is, greatest possible, for each $p\in [1,\infty)$ and each simply or doubly connected domain $\Omega \subset\overline{\mathbb{C}}$ of hyperbolic type. For $c_p(2,\Omega)$ in (22) we have the following result. Proposition 6.5. Let $\Omega \subset {\mathbb{C}}$ be a simply connected domain of hyperbolic type. Then the following hold: (i) if $p\in [1,2]$, then $2^{p-4}/p^p\leqslant c_p(2,\Omega)\leqslant 2^{4-p}/p^p$; (ii) if $p\in [2,\infty)$, then ${1}/{(2p)^p}\leqslant c_p(2,\Omega)\leqslant {2^{p}}/{p^p}$. In the proofs of these results we can use Theorem 6.5 and the following result: for a simply connected domain $\Omega \subset {\mathbb{C}}$ of hyperbolic type the inequalities $R(z,\Omega)/4\leqslant \rho(z,\partial\Omega) \leqslant R(z,\Omega)$, $z\in \Omega$, hold by Koebe’s $1/4$-theorem and the principle of the hyperbolic metric. We deduce these estimates with details only in one case. Let $p\in [1,2]$. Then
$$
\begin{equation*}
\frac{1}{R^{2-p}(z,\Omega)}\leqslant \frac{1}{\rho^{2-p}(z,\partial\Omega)}\,, \quad \frac{1}{R^{2}(z, \Omega)}\geqslant \frac{1}{16 \rho^2(z,\partial\Omega)}\,, \qquad z\in \Omega.
\end{equation*}
\notag
$$
Hence (30) yields the inequality
$$
\begin{equation*}
\iint_{\Omega} \frac{|\nabla u (z)|^p\,dx\,dy}{\rho^{2-p}(z,\partial\Omega)} \geqslant \frac{2^p}{16 p^p} \iint_\Omega\frac{|u(z)|^p\, dx\,dy}{\rho^2(z,\partial\Omega)} \quad \forall\,u\in C_0^1(\Omega).
\end{equation*}
\notag
$$
Since $c_p(2,\Omega)$ is by definition the greatest possible constant in the similar inequality (22), we have $c_p(2,\Omega)\geqslant 2^{p-4}/p^p$. Now for $p\in [1,2]$ we have
$$
\begin{equation*}
\frac{4^{2-p}}{R^{2-p}(z,\Omega)}\geqslant \frac{1}{\rho^{2-p}(z,\partial\Omega)}\,, \quad \frac{1}{R^{2}(z,\Omega)}\leqslant \frac{1}{\rho^2(z,\partial\Omega)}\,, \qquad z\in \Omega.
\end{equation*}
\notag
$$
Therefore, (22) yields
$$
\begin{equation*}
\iint_{\Omega} \frac{|\nabla u (z)|^p\,dx\,dy}{R^{2-p}(z,\Omega)} \geqslant\frac{c_p(2,\Omega)}{4^{2-p}} \iint_\Omega \frac{|u(z)|^p\,dx\,dy}{R^{2}(z,\Omega)} \quad \forall\,u\in C_0^1(\Omega).
\end{equation*}
\notag
$$
As $2^p/p^p$ is the greatest constant in (30), we obtain
$$
\begin{equation*}
\frac{c_p(2, \Omega)}{4^{2-p}}\leqslant \frac{2^p}{ p^p}.
\end{equation*}
\notag
$$
Hence $c_p(2,\Omega)\leqslant 2^{4-p}/p^p$. In doubly connected domains, for $c_p(2,\Omega)$ in (22) we have the following result. Proposition 6.6. Let $\Omega \subset {\mathbb{C}}$ be a doubly connected domain of hyperbolic type, and let $M_0(\Omega)<\infty$. Then the following hold: (i) if $p\in [1,2]$, then
$$
\begin{equation*}
\frac{2^{p}}{p^p(4 M_0(\Omega)+2+2\sqrt{2}\,)^2}\leqslant c_p(2, \Omega)\leqslant \min\biggl\{\frac{2^{p}}{p^p}\,,\frac{1}{2^{-p}M_0(\Omega)^p}\biggr\};
\end{equation*}
\notag
$$
(ii) if $p\in [2,\infty)$, then We justify the lower bounds by following the proof of Proposition 6.5 and using Theorem 6.5 and the following result: for a doubly connected domain $\Omega \subset {\mathbb{C}}$ of hyperbolic type
$$
\begin{equation*}
\frac{R(z,\Omega)}{4M_0(\Omega)+2+2\sqrt{2}} \leqslant \rho(z,\partial\Omega)\leqslant R(z,\Omega), \qquad z\in \Omega.
\end{equation*}
\notag
$$
The left-hand bound was proved in [17]. It is asymptotically sharp as $M(\Omega)\to \infty$ in the following sense: for any sequence of doubly connected domains $\Omega_n \subset \mathbb{C}$ with finite moduli $M(\Omega_n)$ such that $M(\Omega_n)\to \infty$ as $n\to \infty$, the following equality holds (see [17]):
$$
\begin{equation*}
\lim_{n\to \infty}\,\frac{1}{4M_0(\Omega_n)}\, \sup_{z\in \Omega_n}\,\frac{R(z,\Omega_n)}{\rho(z,\partial\Omega_n)}=1.
\end{equation*}
\notag
$$
Now we prove the right-hand bounds for $c_p(2,\Omega)$. Let $p\in [2,\infty)$. Then from (22), in view of the inequality $\rho(z,\partial\Omega)\leqslant R(z,\Omega)$ we obtain
$$
\begin{equation*}
\iint_{\Omega}\frac{|\nabla u(z)|^p\,dx\,dy}{R^{2-p}(z,\Omega)} \geqslant {c_p(2,\Omega)} \iint_\Omega\frac{|u(z)|^p\,dx\,dy}{R^{2}(z,\Omega)}\quad \forall\,u\in C_0^1(\Omega).
\end{equation*}
\notag
$$
As ${2^p}/{p^p}$ is the greatest constant in (30), we have Combining this with the right-hand estimate for $c_p(2,\Omega)$ from Theorem 6.1, we obtain the required result. Note that for $p=2$ the right-hand estimate yields $c_2(2,\Omega)\leqslant 1$. For $p\in [1,2]$ the right-hand bound for $c_p(2,\Omega)$ follows from three facts: the estimate $c_2(2,\Omega)\leqslant 1$, the inequality $c_p(2,\Omega)\leqslant (2/p)^p c_2^{p/2}(2,\Omega)$ (see [9]), and the right-hand estimate for $c_p(2,\Omega)$ from Theorem 6.1. Using right-hand estimates for $c_2(2,\Omega)=c_2(\Omega)$ and the inequalities
$$
\begin{equation*}
\sqrt{C^*_2(2,\Omega)}\geqslant c_2(\Omega)\quad\text{and}\quad A_m(\Omega)\geqslant \bigl((m-1)!\bigr)^2 c_2(\Omega),
\end{equation*}
\notag
$$
we obtain the following propositions. Proposition 6.7. Let $m \in \mathbb{N}$, $m\geqslant 2$, and let $\Omega \subset {\mathbb{C}}$ be a simply connected domain if hyperbolic type. Then for each real function $u\in C_0^\infty(\Omega)$ and Proposition 6.8. Let $m \in \mathbb{N}$, $m\geqslant 2$, and let $\Omega \subset {\mathbb{C}}$ be a doubly connected domain of hyperbolic type. If the Euclidean maximum modulus of $\Omega$ satisfies $M_0(\Omega)<\infty$, then for each real function $u\in C_0^\infty(\Omega)$ and Now we present several integral inequalities in weakly convex domains. We require the scalar product $\nabla u(z)\cdot \nabla \rho(z,\partial\Omega)$ and the inner radius $\rho(\Omega)$ of the domain $\Omega\subset\mathbb{C}$, which is defined by
$$
\begin{equation*}
\rho(\Omega)=\sup_{z\in \Omega}\rho(z,\partial\Omega).
\end{equation*}
\notag
$$
It is obvious that for bounded and some unbounded domains $\Omega\subset\mathbb{C}$ the least upper bound can be replaced by the maximum value; then the inner radius of $\Omega$ is the radius of the maximal disc embedded in $\Omega$. To state the following result we need the constant $\Lambda_2$ introduced in [10]. Consider a solution of Gauss’s hypergeometric equation
$$
\begin{equation*}
\zeta(1-\zeta)\eta''+(1+2\zeta)\eta'-\frac{1}{2}\eta=0
\end{equation*}
\notag
$$
which is holomorphic in $\mathbb{C}\setminus (-\infty,0]$. We use the solution $\eta(\zeta)$ only for real values of $\zeta=\operatorname{Re}\zeta=\xi \in (0,\infty)$. Let $v(\xi):=\eta(1+\xi)$. Let $\Lambda_2$ denote the first positive root of a solution of the Lamb-type transcendental equation It was proved in [10] that such a root does exist and $\Lambda_2\approx 2.49$. The expressions ‘exterior sphere condition’ and ‘exterior ball condition’ have various meanings in the literature. We use the following definition. Definition 6.1. Given a positive number $\lambda$, we say that a domain $\Omega\subset\mathbb{C}$ is $\lambda$-close to convex if $\Omega \ne {\mathbb{C}}$ and for each boundary point $\zeta\in (\partial\Omega) \setminus\{\infty\}$ there exists a point $a_\zeta \in \mathbb{C}\setminus \overline{\Omega}$ such that $|\zeta-a_\zeta|=\lambda$ and the disc $D_\zeta =\{z\in \mathbb{C}\colon |z-a_\zeta| <\lambda\}$ lies in $\mathbb{C} \setminus \overline{\Omega}$. If in this definition the domain $\Omega\subset\mathbb{C}$ is bounded, then $\Omega \ne {\mathbb{C}}$ holds automatically and $(\partial\Omega)\setminus\{\infty\}=\partial\Omega$. It is obvious that if the domain $\Omega\subset\mathbb{C}$ is convex and $\Omega \ne {\mathbb{C}}$, then this domain is $\lambda$-close to a convex one for each $\lambda\in (0,\infty)$. The geometric assumptions about $\Omega\subset\mathbb{C}$ in this definition are related to the conditions of weak convexity in the sense of Efimov–Stechkin and in the sense of Vial (see [67], [88], [146], and [147]). Note that the assumptions on the domain in Definition 6.1 ensure that the set of interior points of $\overline{\Omega}$ coincides with $\Omega$ and, moreover, $\Omega$ is a component of the set of interior points of a set $X\subset\mathbb{R}^n$ which is $r$-weakly convex for $r=\rho(\Omega)/\Lambda_2$ in the sense of Efimov and Stechkin. Theorem 6.6 ([21], 2022). Let $p\in [2,\infty)$, and let $\Omega\subset\mathbb{C}$ be a non-convex domain such that $\rho(\Omega)<\infty$. If $\Omega$ is $\lambda$-close to convex, where $\lambda\geqslant\rho(\Omega)/\Lambda_2$, then We can present a simple example of a measure satisfying the hypotheses of Theorem 6.6. Example 6.1 ([21]). Let $\Omega_2 \subset \mathbb{C}$ be a ‘cross-like’ domain bounded by hyperbolae and defined by Clearly, the unit disc with centre at the origin lies in $\Omega_2$, while any other disc in this domain has radius less than one. Hence the radius of the maximal open disc in $\Omega_2$ is $\rho(\Omega_2)=1$. It is easy to see that $\Omega_2$ is $\lambda$-close to convex for $\lambda=\min R(x)$, where $R(x)$ is the curvature radius of the hyperbola $y=1/x$, $0<x<\infty$, at the point $(x,1/x)$. Because we have $\lambda=\lambda(\Omega_2)=\sqrt{2}$ . It is clear that $\lambda \geqslant \rho(\Omega_2)/ \Lambda_2$, since $\Lambda_2\approx 2.49$. Hence the domain $\Omega=\Omega_2$ satisfies the assumptions of the theorem and the inequalities with sharp constants from Theorem 6.6 hold in it. A number of other Hardy- and Rellich-type inequalities in non-convex domains which are $\lambda$-close to convex were proved by Avkhadiev in [15]. In particular, the following result holds (see [15], the case $d=s=2$ of Theorem 2.1). Theorem 6.7. Let $\Omega\subset\mathbb{C}$ be a domain $\lambda$-close to convex for $\lambda=\lambda_0(\Omega)\in (0,\infty)$. Let $p \in [1,\infty)$ and assume that $\rho(\Omega)\in (0,\infty)$. Then for each function $u\in C_0^{\infty}(\Omega)$ where $\gamma=1+\rho(\Omega)/\lambda_0(\Omega)$. Note the following useful estimate established in [22]:
$$
\begin{equation*}
e^{2\pi M_0(\Omega)}\leqslant 1+\frac{\rho(\Omega)}{\lambda_0(\Omega)}\,,
\end{equation*}
\notag
$$
where $\Omega\subset\mathbb{C}$ is a domain $\lambda$-close to convex for $\lambda=\lambda_0(\Omega)\in (0,\infty)$. We also mention on the way that in [22] the reader can find some applications of Hardy-type inequalities to estimates for torsional rigidity in the sense of Saint-Venant and the first eigenvalue of the Dirichlet problem for Laplace’s equation.
7. On a problem of ProtasovIn this section we present a result based on Hardy’s classical inequality for sequences. Let $c_0,c_1,c_2,\ldots$ be a sequence of complex numbers. Then Hardy’s classical inequality can be written as
$$
\begin{equation*}
\sum_{n=0}^\infty\biggl|\frac{c_0+c_1+\cdots+c_n}{n+1}\biggr|^p \leqslant \biggl(\frac{p}{p-1}\biggr)^p\sum_{n=0}^\infty|c_n|^p, \qquad p>1.
\end{equation*}
\notag
$$
For $p=2$ this is equivalent to the fact that the Cesàro operator ${\mathcal C}f$ introduced in § 3 is bounded in the Hardy space $H_2$ and has norm 2. Let $p_n$ be an algebraic polynomial of degree $n$ with poles at points $z_1,\dots,z_n$. By a simple partial fraction we mean the logarithmic derivative of $p_n$:
$$
\begin{equation*}
g_n(t)=\frac{f_n'(t)}{f_n(t)}=\sum_{k=1}^n\frac{1}{t-z_k}\,.
\end{equation*}
\notag
$$
These fractions have many remarkable properties and a number of physical interpretations, for example, in electrostatics. The reader can find a comprehensive history of simple partial fractions in the paper [45] by Danchenko, Komarov, and Chunaev. We let $z_k=x_k+i y_k \in \mathbb{C}\setminus\mathbb{R}$, $1 \leqslant k \leqslant n$. In [133] Protasov investigated the convergence of simple partial fractions. Using some non-trivial properties of the Hilbert transform he showed that if
$$
\begin{equation}
\sum_{k=1}^\infty \frac{1}{|y_k|^{1/q}} < \infty, \qquad \frac{1}{p}+\frac{1}{q}=1,
\end{equation}
\tag{31}
$$
then the function series converges in $L_p(\mathbb{R})$. Protasov stated a problem: find necessary and sufficient conditions for the convergence of $g_\infty$, and showed that when all the $z_k$ lie in the sector $\{|z| \leqslant C|y|\}$ for a fixed $C$, it is necessary for the convergence of $g_\infty$ that
$$
\begin{equation}
\sum_{k=1}^\infty \frac{1}{|y_k|^{1/q+\varepsilon}} < \infty.
\end{equation}
\tag{32}
$$
We see that the sufficient condition (31) is quite close to the necessary condition (32). However, there is a gap between these conditions which was closed in [90], where a sufficient condition for convergence of type (31) was obtained, which turnes out to be also necessary in the case when all poles $z_n$ lie in a sector $\{|z| \leqslant C|y|\}$ for a fixed $C$. This condition has the form
$$
\begin{equation}
\sum_{k=1}^\infty \frac{k^{p-1}}{|y_k|^{p-1}} < \infty.
\end{equation}
\tag{33}
$$
Using Hardy’s inequality for sequences mentioned above, the following result was proved in [90]. Theorem 7.1. Let $p>1$. If condition (33) holds, then the series is convergent in $L_p(\mathbb{R})$. Conversely, if this series is convergent in $L_p(\mathbb{R})$, the sequence $|y_n|$ is arranged in increasing order, and $|z_k| \leqslant C|y_k|$, then (33) holds. It should be noted that, although condition (33) for convergence is close to (31), it is not a consequence of it. A suitable example is easy to give. 8. Moduli of quadrilaterals and doubly connected domainsConformal moduli play an important role in geometric function theory. We start with the moduli of so-called quadrilaterals. A quadrilateral $\boldsymbol{Q}=(Q;z_1,z_2,z_3,z_4)$ is a Jordan domain $Q$ on the Riemann sphere with four points $z_1$, $z_2$, $z_3$, and $z_4$ marked on its boundary; we assume that the index $j$ of $z_j$ increases in the positive direction of the boundary $\partial Q$. The points $z_j$ are called vertices of the quadrilateral $\boldsymbol{Q}$. The conformal modulus of $\boldsymbol{Q}$ can be defined in several equivalent ways. The first is to use conformal mappings. By Riemann’s classical theorem there exists a conformal mapping $f$ of $Q$ onto a rectangle $[0,1]\times [0,h]$, $h>0$, that takes $z_1$, $z_2$, $z_3$, and $z_4$ to $0$, $1$, $1+ih$, and $ih$. It is easy to show that $h$ is uniquely determined; it is called the conformal modulus of $\boldsymbol{Q}$. We express this by Another way to define a modulus is to use the concept of the extremal length $\lambda(\Gamma)$ of the family of curves $\Gamma$ (for instance, see [3]). If $\Gamma$ is the family of curves in $Q$ connecting the sides $z_1z_2$ and $z_3z_4$ of $\boldsymbol{Q}$, then $\operatorname{Mod}(\boldsymbol{Q})=\lambda(\Gamma)$. On the other hand, if we consider the family $\Gamma_1$ of curves connecting the sides $z_2z_3$ and $z_1z_4$ in $Q$, then $\operatorname{Mod}(\boldsymbol{Q})=1/\lambda(\Gamma_1)$. Finally,
$$
\begin{equation*}
\operatorname{Mod}(\boldsymbol{Q})= \biggl(\inf_u\iint_Q|\nabla u|^2\,dx\,dy\biggr)^{-1},
\end{equation*}
\notag
$$
where the infimum is taken over all smooth functions $u$ that are continuous in $\overline{Q}$, vanish on the boundary arc $z_1z_2$ and are equal to $1$ on $z_3z_4$; it is well known that this infimum is in fact a minimum: it is attained at a harmonic function. For a quadrilateral $\boldsymbol{Q}=(Q;z_1,z_2,z_3,z_4)$ we can define the conjugate quadrilateral $\boldsymbol{Q}^*=(Q;z_2,z_3,z_4,z_1)$. It is obvious that $\operatorname{Mod}(\boldsymbol{Q}^*)= 1/\operatorname{Mod}(\boldsymbol{Q})$. Let $Q$ be a bounded Jordan domain in $\mathbb{C}$, let $L=\partial Q$, and let $z_1$, $z_2$, $z_3$, and $z_4$ be points on $L$ satisfying the above conditions. Consider the corresponding quadrilateral $\boldsymbol{Q}=(Q;z_1,z_2,z_3,z_4)$. We call the quantity $\operatorname{Mod}(Q;z_1,z_2,z_3,z_4)$ the inner modulus. If we consider the quadrilateral $\boldsymbol{Q}^{\rm c}:=(Q^{\rm c};z_4,z_3,z_2,z_1)$, where $Q^{\rm c}$ is the complement to $Q$ on the Riemann sphere, then it is natural to call $\operatorname{Mod}(\boldsymbol{Q}^{\rm c})$ the outer modulus. Conformal moduli are important for geometric function theory and its applications. They are invariant under conformal mappings and quasi-invariant under quasiconformal ones. More precisely, if $f$ is a $K$-quasiconformal mapping of the Jordan domain $Q$ onto a Jordan domain $D$, such that four points $z_1$, $z_2$, $z_3$, and $z_4$ on the boundary of $Q$ are taken to four points $\zeta_1$, $\zeta_2$, $\zeta_3$, and $\zeta_4$ on the boundary of $D$, then for the quadrilaterals $\boldsymbol{Q}=(Q;z_1,z_2,z_3,z_4)$ and $\boldsymbol{D}=(D;\zeta_1,\zeta_2,\zeta_3,\zeta_4)$ we have the inequality
$$
\begin{equation*}
K^{-1}\operatorname{Mod}(\boldsymbol{Q})\leqslant \operatorname{Mod}(\boldsymbol{D})\leqslant K\operatorname{Mod}(\boldsymbol{Q}).
\end{equation*}
\notag
$$
This property opens way to the use of moduli in deriving various estimates for conformal and quasiconformal mappings. The conformal moduli of quadrilaterals are closely connected with the conformal moduli of doubly connected domains. If $G$ is a doubly connected domain with non-degenerate boundary then it can be mapped conformally onto an annulus $1<|\zeta|<q$ (for instance, see [74]). The number $q\in(1,\infty)$ is independent of the choice of the conformal mapping, and the modulus of a doubly connected domain $G$ is by definition Just as for quadrilaterals, there are also other characterizations of the modulus of a doubly connected domain, which can be taken as its definitions. For example, we can show that $\operatorname{Mod}(G)=\lambda(\Gamma)$, where $\lambda(\Gamma)$ is the extremal length of the family $\Gamma$ of curves connecting boundary components of $G$. In addition, $\operatorname{Mod}(G)=1/\lambda(\Gamma')$, where $\lambda(\Gamma')$ is the extremal length of the family $\Gamma'$ of curves separating the boundary components of $G$. Finally,
$$
\begin{equation*}
\operatorname{Mod}(G)^{-1}= \biggl(\,\inf_v\iint_G|\nabla v|^2\,dx\,dy\biggr)^{-1},
\end{equation*}
\notag
$$
where the infimum is taken over all smooth functions $v$ in $G$ that vanish on one boundary component of $G$ and are equal to one on the other component. This equality can also be written as where $\operatorname{Cap}(G)$ is the capacity of the condenser with field $G$ and plates coinciding with the connected components of the complement to $G$. Note that this definition in terms of capacity works for bounded domains in the plane, while in the case of unbounded domains we can use conformal invariance and, in place of $G$, consider a conformally equivalent bounded domain. The conformal moduli of doubly connected domains are quasi-invariant under quasiconformal mappings: if $f$ is a $K$-quasiconformal mapping of the domain $Q$ onto $f(Q)$, then
$$
\begin{equation*}
K^{-1}\operatorname{Mod}(Q)\leqslant \operatorname{Mod}(f(Q))\leqslant K\operatorname{Mod}(Q).
\end{equation*}
\notag
$$
The properties of conformal moduli of doubly connected domains were rather well exposed in Künau’s survey [98] (also see [123]). As concerns the capacities of condensers and their applications to geometric function theory, see, for example, [56]. Let us dwell on the problem stated by Vuorinen in 2004, at a conference in Volgograd. Let $G$ be a bounded doubly connected domain in the plane. Consider a quasiconformal mapping $f_H$ of the plane onto itself that is the dilation with coefficient $H>0$ along the $x$-axis: What is the behaviour of the conformal modulus of the domain $G_H=f_H(G)$ as $H\to \infty$? In particular, what occurs when $G$ is an annulus or the difference of two homothetic squares?First we look at the analogous problem of dilating a quadrilateral $(Q;z_1,z_2, z_3,z_4)$ such that
$$
\begin{equation}
Q=\{(x,y)\in\mathbb{R}^2\colon f(x)\leqslant y\leqslant g(x),\, a\leqslant x \leqslant b\},
\end{equation}
\tag{34}
$$
where $f$ and $g$ are continuous functions on the interval $[a,b]$ such that $f(x) < g(x)$ for $x\in [a,b]$; the vertices of this quadrilateral are
$$
\begin{equation}
z_1=a+ig(a), \quad z_2=a+if(a),\quad z_3=b+if(b), \quad\text{and}\quad z_4=b+ig(b).
\end{equation}
\tag{35}
$$
Set $\theta(x)=g(x)-f(x)$ and $\varphi(x)=[g(x)+f(x)]/2$. We start by recalling classical results due to Ahlfors (see [3], Chap. 4, § 5) and Warshawski (for instance, see [68], Chap. 2, § 5, and [110], Chap. 7, § 1) on estimates for the conformal moduli of quadrilaterals. Theorem 8.1 (Ahlfors–Warshawski). Let $f$ and $g$ be continuously differentiable functions. Then the conformal modulus of the quadrilateral $\boldsymbol{Q}=(Q;z_1,z_2,z_3,z_4)$ satisfies the inequalities Note that the lower estimate in Theorem 8.1 is due to Ahlfors, and it also holds for non-smooth functions $f$ and $g$. The upper bound is due to Warshawski. Now consider the quadrilateral
$$
\begin{equation*}
\boldsymbol{Q}_H:=(f_H(Q);z_{1H},z_{2H},z_{3H},z_{4H}),
\end{equation*}
\notag
$$
where the domain $Q$ was defined in (34) and $z_{jH}=f_H(z_j)$, $1\leqslant j\leqslant 4$. Theorem 8.1 yields the following result. Theorem 8.2. The conformal modulus $\boldsymbol{Q}_H$ has the following asymptotic behaviour: Moreover, $\operatorname{Mod}(\boldsymbol{Q}_H)\geqslant cH$ and $\operatorname{Mod}(\boldsymbol{Q}_H)-cH=O(H^{-1})$, $H\to\infty$. Note that in Theorem 8.2 the functions $f$ and $g$ are not assumed to be smooth. For the first time it was in fact established in [46]; a shorter proof using Theorem 8.1 was given in [120]. We turn to the solution of the Vuorinen problem. We consider doubly connected domains $G$ of the following form. The boundary of $G$ satisfies the conditions where
$$
\begin{equation*}
\begin{aligned} \, \Gamma_1&=\{x+iy\colon y=f_1(x),\, a\leqslant x\leqslant b\}\cup\newline\{x+iy\colon y=f_2(x), \, a\leqslant x\leqslant b\} \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, \Gamma_2&=\{x+iy\colon y=g_1(x), \, c\leqslant x\leqslant d\}\cup\newline\{x+iy\colon y=g_2(x), \, c\leqslant x\leqslant d\}, \end{aligned}
\end{equation*}
\notag
$$
where 1) $-\infty<a<c<d<b<\infty$; 2) $g_1(x)<f_1(x)$ and $f_2(x)<g_2(x)$ for $c\leqslant x\leqslant d$; 3) $g_2(x)<g_1(x)$ for $c<x<d$; 4) $f_2(x)<f_1(x)$ for $a<x<b$; 5) $f_2(a)=f_1(a)$, $f_2(b)=f_1(b)$, $g_2(c)=g_1(c)$, for $g_2(d)=g_1(d)$. Let $G_H$ be obtained from the domain $G$ by means of the dilation $f_H$. We partition $G_H$ into three parts by vertical lines $x=cH$ and $x=dH$. Two of these parts are the quadrilaterals
$$
\begin{equation*}
\begin{aligned} \, G_{H1}&=\biggl\{(x,y)\colon g_1\biggl(\frac{x}{H}\biggr)<y<f_1\biggl(\frac{x}{H}\biggr),Hc<x<Hd\biggr\} \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, G_{H2}&=\biggl\{(x,y)\colon f_2\biggl(\frac{x}{H}\biggr)<y<g_2\biggl(\frac{x}{H}\biggr),Hc<x<Hd\biggr\}. \end{aligned}
\end{equation*}
\notag
$$
It turns out that when $G_H$ is mapped onto an annulus $1<|z|<q_H$, the quantity $q_H$ tends to one as $H\to\infty$. Moreover, by modifying Rado’s theorem on a criterion for the uniform convergence of conformal mappings we can show that the images of the line segments contract to points. Taking account of the conformal invariance of the modulus, this implies that the conformal modulus of $G_H$ has the same asymptotic behaviour as the sum of the conformal moduli of the quadrilaterals $G_{H1}$ and $G_{H2}$. The asymptotics of their moduli can be found using Theorem 8.2. This yields the following theorem, which was proved by Dautova and Nasyrov [47]. Note some partial cases of Theorem 8.3 obtained in [46]. Using Theorem 8.3 one can find the asymptotic behaviour of the modulus of an annulus under dilations with coefficient $H\to\infty$. Note that under such dilations the boundary circles become non-confocal ellipses, so one cannot write out the conformal mapping of the dilated annulus onto a circular annulus explicitly using known elementary and special functions. Theorem 8.4. If $D=\{z\in \mathbb{C}\colon r<|z|<R\}$, then where We find an estimate for the error in the case when the doubly connected domain has the form $D=\{(x,y)\colon 1<|x|+|y|<k\}$, $k>1$, so that it is the difference of two squares with vertices on coordinate axes. Then
$$
\begin{equation}
D_H=\biggl\{(x,y)\colon 1<\biggl|\frac{x}{H}\biggr|+|y|<k\biggr\}.
\end{equation}
\tag{36}
$$
By Theorem 8.3
$$
\begin{equation*}
\operatorname{Mod}(D_H)\sim \frac{k-1}{4H}\,,\qquad H \to \infty.
\end{equation*}
\notag
$$
We estimate the accuracy of this asymptotic formula. For the modulus of the rhombus-shaped doubly connected domain $D_H$ defined in (36) we have an estimate due to Dautova:
$$
\begin{equation*}
\frac{k-1}{4H}-\frac{\log 2}{2\pi}\,\frac{(k-1)^2}{H^2}+ O\biggl(\frac{1}{H^3}\biggr)\leqslant \operatorname{Mod}(D_H)\leqslant \frac{k-1}{4H}\,,\qquad H\to\infty.
\end{equation*}
\notag
$$
Theorem 8.3 can be extended to domains in wider classes. For example, it holds when the outer boundary component is unbounded (the function $g$ is defined on the whole axis and can take the infinite value) and also in the case when the boundary components are not the graphs of single-valued functions and can even not be curves, but rather continua of a fairly arbitrary shape. In [120] and [122] the authors considered the Vuorinen problem for doubly connected domains $\Omega$ containing the point at infinity in their interiors. Also assume that the boundary of $\Omega$ is formed by the graphs of continuous functions: the functions $g_1$ and $f_1$ are defined on $[a,b]$ so that $f_1(x)\leqslant g_1(x)$, $x\in[a,b]$; and $g_2$ and $f_2$ are defined on $[c,d]\subset[a,b]$ so that $g_2(x)\leqslant f_2(x)<f_1(x)$, $x\in[c,d]$. In addition, $f_1(a)=g_1(a)$, $f_1(b)=g_1(b)$ and $f_2(c)=g_2(c)$, $f_2(d)=g_2(d)$. Nguen and Nasyrov [120], [122] established the following analogue of Theorem 8.3. For the proof of Theorem 8.5 the domain $\Omega_H$ is partitioned into two parts by the vertical line segments of the lines $x=cH$ and $x=dH$ that connect the boundary components of $\Omega_H$. The bounded part is a quadrilateral, whose modulus can be investigated with a reference to Theorem 8.1. As concerns the unbounded component, the (outer) modulus of the corresponding quadrilateral is of order $\log H$ as $H\to\infty$, so that this unbounded component does not affect the behaviour of the leading term of the asymptotic of the conformal modulus. An interesting problem arises in this connection: describe the asymptotic behaviour of the outer modulus under dilations. Consider the quadrilateral $\boldsymbol{Q}=(Q; z_1, z_2, z_3, z_4)$, where the domain $Q$ and its vertices are given by (34) and (35). Let the quadrilateral $\boldsymbol{Q}_H$ be obtained from $\boldsymbol{Q}$ by a dilation with coefficient $H>0$, and let $\boldsymbol{Q}^{\rm c}_H$ be the quadrilateral corresponding to its complement. Task 8.1. Describe the asymptotic behaviour of the conformal modulus of $\boldsymbol{Q}^{\rm c}_H$ as $H\to\infty$. It is easy to show that $\operatorname{Mod}(\boldsymbol{Q}^{\rm c}_H)=O(\log H)$ as $H\to\infty$. Consider the special case when $Q$ is the rectangle $\Pi_{ab}:=[0,a]\times [0,b]$. Then there is an elegant formula linking the inner and outer conformal moduli, which is due to Duren and Pfaltzgraff [64]; in comparison to [64] our numbering of vertices is shifted by one, as in (35). While the outer modulus of the rectangle is $2\mathcal{K}(r)/\mathcal{K}'(r)$, its inner modulus, that is, the ratio $a/b$ of its sides, is $\psi(r)$, where
$$
\begin{equation*}
\psi(r):=\frac{2[\mathcal{E}(r)-(1-r)\mathcal{K}(r)]} {\mathcal{E}'(r)-r\mathcal{K}'(r)}\,.
\end{equation*}
\notag
$$
Here $\mathcal{K}(r)$ and $\mathcal{E}(r)$ are complete elliptic integrals of the first and second kind; furthermore, $\mathcal{K}'(r):=\mathcal{K}(r')$ and $\mathcal{E}'(r):=\mathcal{E}(r')$, where $r'=\sqrt{1-r^2}$ . Using this result, Vuorinen and Zhang [148] established in fact an estimate for the outer modulus $\operatorname{ExtMod}(\Pi_{ab})$ of the rectangle $\Pi_{ab}$:
$$
\begin{equation*}
\begin{aligned} \, &\frac{2}{\pi}\biggl[1-\biggl(1+ \sqrt{\frac{4}{\pi}\,\frac{a}{b}}\,\biggr)^{-1}\biggr] \log\biggl[2\biggl(1+\sqrt{\frac{4}{\pi}\,\frac{a}{b}}\,\biggr)\biggr] \\ &\qquad<\operatorname{ExtMod}(\Pi_{ab})< \frac{2}{\pi}\,\log\biggl[2\biggl(1+\sqrt{\pi\frac{a}{b}}\,\biggr)\biggr]. \end{aligned}
\end{equation*}
\notag
$$
It immediately follows from this estimate that when $\Pi_{ab}$ is dilated with coefficient $H$, its outer modulus is equivalent to the quantity $\pi^{-1}\log H$, independent of $a/b$. Thus we can make the following conjecture. Conjecture 8.1. For each quadrilateral $\boldsymbol{Q}=(Q;z_1,z_2,z_3,z_4)$, defined by (34) and (35), its outer conformal modulus has the asymptotic behaviour 9. Reduced moduli, Robin capacities, and their applicationsAlong with conformal moduli of quadrilaterals and doubly connected domains, the so-called reduced conformal moduli are also of considerable interest. Let $D$ be a simply connected domain with non-degenerate boundary on the extended complex plane, and let $z_0$ be a point in $D$. From $D$ we remove a disc of sufficiently small radius $\varepsilon>0$ with centre $z_0$. This doubly connected domain $D_\varepsilon$ has a conformal modulus $m(\varepsilon)$, which tends to infinity as $\varepsilon\to0$. One can show that a finite limit
$$
\begin{equation*}
r(D,z_0):=\lim_{\varepsilon\to 0}\biggl(m(\varepsilon)- \frac{1}{2\pi}\log \frac{1}{\varepsilon}\biggr)
\end{equation*}
\notag
$$
exists; it is called the reduced (conformal) modulus of $D$ at $z_0$. Note that the reduced modulus of a simply connected domain $D$ at a point is equal to the conformal radius of $D$ at this point. We can also consider unbounded domains $D$ containing the point at infinity in their interior. From such domain we remove the exterior of a disc of radius $R$, where $R$ is sufficiently large, and the reduced modulus is defined by
$$
\begin{equation*}
r(D,\infty):=\lim_{R\to \infty}\biggl(m(R)-\frac{1}{2\pi}\log R\biggr),
\end{equation*}
\notag
$$
where $m(R)$ is the conformal modulus of the domain $D_R=D\setminus \{|z|\geqslant R\}$. We can define such a quantity not only for simply connected domains, but also for domains of any connectivity; in this case, in place of the modulus of a doubly connected domain, we take as $m(\varepsilon)$ the extremal length of the family of curves connecting the circle $|z-z_0|=\varepsilon$ with the boundary of $D$ within $D_\varepsilon$. One can also define a reduced modulus using the capacity of the condenser with field $D _\varepsilon$ and plates equal to the disc $|z-z_0|<\varepsilon$ and $D^{\rm c}=\overline{\mathbb{C}}\setminus D$. There are various generalizations of the concept of reduced conformal modulus. For example, Dubinin introduced reduced moduli when, in place of a neighborhood of a point, neighbourhoods of several points are removed so that the rate of convergence to zero of the radii of these neighbourhoods can be different (see [54], [56], [62], and [59]). The neighbourhoods removed are even non necessarily circular, but can be domains bounded by smooth curves and contracting to points [60]. Also of interest are boundary reduced moduli (see [141]), when a neighbourhood of a boundary point is removed. Such moduli play a major part in estimates for univalent and non-univalent functions, for instance, polynomials and rational functions. Note yet another generalization of a reduced conformal modulus, which is related to so-called Robin capacities. Let us define this concept in the case of the point at infinity. Let $D$ be a Jordan domain in $\overline{\mathbb{C}}$ containing a point at infinity. Let $A$ be a connected non-degenerate subset of the boundary $\partial D$. For sufficiently large $R$ the boundary $\partial D$ lies in the disc $K_R$ of radius $R$ with centre at the origin. Let $\Gamma_R$ be the family of curves in $D\cap K_R$ that connect $A$ with the circle $C_R=\{z\colon |z|=R\}$, and let $\lambda(\Gamma_R)$ be the extremal length of $\Gamma_R$. Then the limit
$$
\begin{equation*}
\mu(A;D):=\lim_{R\to\infty}\biggl(\lambda(\Gamma_R)- \frac{1}{2\pi}\log R\biggr)
\end{equation*}
\notag
$$
exists. We call the quantity $\mu(A;D)$ the reduced modulus of $A$ with respect to $D$, and we call the constant the Robin capacity of $A$ with respect to $D$ (for instance, see [64]–[66]). It is clear that if $A=\partial D$, then $\sigma(A;D)$ is equal to the Robin capacity, as well as to the Chebyshev constant and transfinite diameter of $D^{\rm c}$ (for instance, see [74]). We can also define Robin capacities for domains of more general form, which are not necessarily Jordan. Also the set $A$ is not necessarily connected. However, we assume in what follows that the above assumptions are fulfilled. An interesting hydromechanical interpretation of Robin capacities was given in [112] in the case of infinitesimally thin airfoils modelled by a Jordan arc $\gamma$; the leading critical point and trailing edge of the flow are the endpoints of the arc. Let $\gamma^+$ and $\gamma^-$ denote the upper and lower sides of the cut along $\gamma$. Consider the conformal mapping $z=f(\zeta)$ of the exterior of the unit disc $E=\{|z|<1\}$ onto the complement to $\gamma$ such that $f(\infty)=\infty$, $f(1)=z_\gamma^{(1)}$, and $f(e^{i \varphi})=z_\gamma^{(2)}$, where $z_\gamma^{(1)}$ and $z_\gamma^{(2)}$ are the endpoints of $\gamma$, the part $\{e^{i\theta},\theta\in [0,\varphi]\}$ corresponds to $\gamma^+$, and its complement corresponds to $\gamma^-$. Nasyrov established the following theorem. Theorem 9.1. The Robin capacities $\sigma(\gamma^+)$ and $\sigma(\gamma^-)$ with respect to $ \mathbb{C}\setminus\gamma$ are equal to respectively, where $d$ is the transfinite diameter of $\gamma$. Furthermore, where $P$ is the dimensionless aerodynamical lift of the arc $\gamma$. It was noted in [113] that Theorem 9.1 also holds for arbitrary not necessarily infinitesimally thin airfoils. Variations of Robin capacities. We describe the variation of the Robin capacity under smooth variations of the boundary of the domain. Let $D$ be a Jordan domain in $\overline{\mathbb{C}}$ containing the point at infinity, and let $\partial D$ contain an arc $A$. Let $\eta$ be a closed subset of $A$, and let the family of domains $D_t$, $0<t\leqslant t_0$, be obtained from $D$ by ‘pushing’ $\eta$ inside $D$, so that the boundary $\partial D_t$ is obtained from $\partial D$ by replacing $\eta$ by an arc $\eta_t$ that lies in $D$ apart from its endpoints. Let $A_t$ denote the part of the boundary of $D_t$ obtained from $A$ by replacing $\eta$ by $\eta_t$. Let $\Omega_t$ be the finite domain bounded by $\eta$ and $\eta_t$. Let $g$ denote the conformal mapping of $D$ onto a domain $G(\infty,x_0)$, where $G(R,x_0)$ is the annulus $G(R)=\{z\in \mathbb{C}\colon 1<|z|<R\}$ with radial cut from $1$ to a point $x_0$, such that $\infty$ goes to $\infty$, the arc $A$ is taken to the unit circle, and its complement to the cut $[0,x_0]$. When the arcs $\eta_t$ and $\eta$ are smooth, we say that they satisfy condition $\mathbf{(\widetilde{R})}$ if for each $\varepsilon>0$ there exists $t_\varepsilon$ such that for each $t\leqslant t_\varepsilon$ we can define a homeomorphic correspondence $\psi_t$ between the points of $\eta_t$ and $\eta$ such that the corresponding points have distance less than $\varepsilon$ and the slope angles of the tangents at these points are less than $\varepsilon$. The following result was established in [113]. Theorem 9.2. The following inequalities hold for reduced moduli and the corresponding Robin capacities: By use of Theorems 9.1 and 9.2, in [113] a problem was solved which generalizes Lavrentiev’s well-known problem on an arc with the greatest aerodynamical lift [101]. Theorem 9.3. Among all smooth arcs of length $l$ with curvature satisfying where $s$ is the natural parameter on the arc and $\psi$ is a non-negative continuous function such that $K_0 l\leqslant c_0$, where an arc with curvature $K(s)\equiv \psi(s)$, $s\in[0,l]$, has the greatest aerodynamical lift. Note that the constant $c_0$ in Theorem 9.3 gives in fact an estimate for the variation of the slope angle of the tangent to the arc in the motion along this arc. Also note that Lavrentiev himself considered only constat majorants $\psi$, and in [101] he established his result with the constant Thus, in Theorem 9.3 this costant is increased with coefficient greater than $6.35$, and the result of the theorem holds for convex arcs with variation of the angle at most $17^\circ$.10. One-parameter families and the dynamics of critical valuesIn the theory of univalent functions Loewner’s parametric method is of importance. It consists in the investigation of a family of univalent conformal mappings $f(z,t)$ in a domain $D$ of the complex variable $z$ (a unit disc, a half-plane, or another domain) and depending on the real parameter $t$. For fixed $t$ the function $f(z,t)$ maps $D$ onto a domain $G(t)$ of complex variable, where $G(t)$ depends on $t$ monotonically. The family $f(z,t)$ is differentiable with respect to $t$ everywhere or almost everywhere on the interval under consideration, and the partial differential equation
$$
\begin{equation}
\frac{\partial f(z,t)}{\partial t}= \frac{\partial f(z,t)}{\partial z}\,h(z,t)
\end{equation}
\tag{37}
$$
is satisfied, where $h(z,t)$ is a holomorphic or meromorphic function of $z$; its properties with respect to $t$ depend on the smoothness of the family $f(z,t)$. Note that an equation of the form (37) is often called a Loewner–Kufarev equation; Loewner himself considered such a question for $h(z,t)$ of more special form. It is also of interest to consider one-parameter families of non-univalent mappings, for instance, polynomials or rational functions. Such families can be regarded on the whole complex plane. In this case critical points, that is, points where $\partial f(z,t)/\partial z\kern-1pt=\kern-1pt 0$, and the images of these points, that is, critical values, play in fact the role of boundary points, along with the poles of these functions. Here are some concrete results. Rational functions. Each rational function $R$ can be expressed in the form
$$
\begin{equation*}
w=R(z)=C\int_{a_1}^z\frac{\prod_{k=1}^M(\zeta-a_k)^{m_k-1}\,d\zeta} {\prod_{j=1}^N(\zeta-b_j)^{n_j+1}}+A_1,
\end{equation*}
\notag
$$
where the $a_k$ are the critical values of $w$ of multiplicity $m_k$, the $b_j$ are poles of multiplicity $b_j$, $C\ne 0$ is a complex constant, and $A_1=R(a_1)$ is a critical value of $R$ corresponding to the critical point $a_1$. Set $A_k=R(a_k)$, $1\leqslant k\leqslant M$. Next we consider one-parameter families of rational functions of the form
$$
\begin{equation}
R(z)=R(z,t)=\int_{a_1}^z\frac{\prod_{k=1}^M(\zeta-a_k)^{m_k-1}\,d\zeta} {\prod_{j=1}^N(\zeta-b_j)^{n_j+1}}\,.
\end{equation}
\tag{38}
$$
The problem is to find an equation of the form (37) which holds for the family $R(z,t)$, provided that we know the laws $A_j=A_j(t)$ governing their critical values. Note that, geometrically, this means that we have a one-parameter family of Riemann surfaces over the Riemann sphere and are looking for a differential equation for the uniformizing functions. This equation can be used in the problem of finding an approximate uniformizing function for a fixed Riemann surface, given as a branched cover of the Riemann sphere. Set $m:=\sum_{k=1}^M(m_k-1)$ and $n:=\sum_{j=1}^N(n_j+1)$. We assume that $l:=m-n\geqslant0$; this means that $R(\infty,t)=\infty$. Then at infinity we have a Laurent expansion
$$
\begin{equation*}
R(z,t)=\frac{1}{l+1}z^{l+1}+\frac{\alpha_l}{l}z^l+\cdots,\quad\text{where}\quad \alpha_l=\sum_{k=1}^M(m_k-1)a_k-\sum_{j=1}^N{(n_j+1)b_j}.
\end{equation*}
\notag
$$
In addition, we assume that the critical points $a_k=a_k(t)$ and poles $b_k=b_k(t)$ satisfy the relation $\alpha_l(t)=0$, and for $l=0$ we also assume that for fixed $t$
$$
\begin{equation*}
R(z,t)=z+O\biggl(\frac{1}{z}\biggr), \qquad z\to\infty.
\end{equation*}
\notag
$$
Note that using these conditions does not significantly restrict the generality, because we can satisfy them by means of linear transformation in the $z$-plane and target plane. Let the critical values $A_l=A_l(t)$ depend smoothly on $t$ (by (38) we have $A_1\equiv 0$). We denote the $t$-derivatives of these functions by $\dot{A}_l$ (in what follows we also denote all $t$-derivatives by dots over the corresponding symbols). Theorem 10.1 ([115] and [117]). The family of functions $R(z,t)$ satisfies the differential equation where $P_{l,j}$ is the Taylor polynomial of degree $j$ of the function at the point $a_l$. Set
$$
\begin{equation*}
G_{kl}(z)=\frac{H_k(z)}{z-a_l}\quad\text{and} \quad I_{kj}(z)=\frac{H_k(z)}{z-b_j}\,.
\end{equation*}
\notag
$$
Theorem 10.1 yields the following. Theorem 10.2 ([117]). The critical points $a_l$, $1\leqslant l\leqslant M$, and poles $b_j$, $1\leqslant j\leqslant M$, of the functions $R(z,t)$ satisfy the system of differential equations We describe how we can use Theorem 10.2 to find approximately the rational function uniformizing a prescribed branched cover $\mathcal{R}_1$. To solve this problem we must know some Riemann surface $\mathcal{R}_0$ with the same branching type and with a known uniformizing function, so that we know the positions of critical points and critical values. Next we must find a smooth path $\mathcal{C}$ in the space of branched covers with this branching type of the sphere that connects the surfaces $\mathcal{R}_0$ and $\mathcal{R}_1$. This path defines the motion of the critical values $A_j$. Next we solve the Cauchy problem for system (39) with the initial data corresponding to $\mathcal{R}_0$. At the terminal moment of time we obtain the required values of the critical points $a_l$ and poles $b_j$ for the surface $\mathcal{R}_1$. Note that, in our opinion, it is an interesting problem to find the ‘most economical’ path $\mathcal{C}$ connecting two surfaces (two branched covers of the sphere with the same branching type). Note that, given the projections $A_1$, $A_2,\dots,A_M$ and a branching type (including the branching over the point at infinity), we cannot in general determine the branched cover uniquely. This was originally discovered by Hurwitz; he put forward the problem of finding the number of non-equivalent covers with prescribed branching type and sketched some approaches to this problem [85], [86] (also see [149]). Subsequently, a significant contribution to the solution of Hurwitz problem was made by Mednykh [108], [109], who derived effective formulae for the calculation of the number of inequivalent covers. The reader can find a description of several ways to the investigation of this problem in [114]. In [99] and the monograph [100] some geometric approaches were described, which are based on the theory of vector bundles. Also note that Theorems 10.1 and 10.2 can be used in problems related to polynomials and rational functions, in particular, in the Smale problem, which we consider in the last section, and in some related questions. Elliptic functions. One can also consider the analogous problem of describing, by means of differential equations, one-parameter families of elliptic functions with periods $\omega_1$ and $\omega_2$ that uniformize a prescribed family of complex tori, that is, Riemann surfaces covering the Riemann sphere with branchings. Here the periods $\omega_1$ and $\omega_2$ depend on the parameter $t$ in general. This problem was considered by Nasyrov [116], [118], [119]. Consider the family of functions of the form
$$
\begin{equation}
f(z)=f(z,t)=c \int_{a_0}^z\frac{\prod_{j=0}^N\sigma^{m_j}(\xi-a_j)} {\prod_{l=0}^P\sigma^{n_l}(\xi-b_l)}\,d\xi+A_0,
\end{equation}
\tag{40}
$$
where $\sigma(z)=\sigma(z;\omega_1,\omega_2)$ is the Weierstrass elliptic $\sigma$-function with periods $\omega_1$ and $\omega_2$; the points $a_0,a_1,\dots,a_N$ are pairwise different and are critical points of $f$; the integers $m_0+1,m_1+1,\dots,m_N+1$ are their multiplicities; the pairwise different points $b_0,\ldots,b_P$ are poles of multiplicities $n_0-1,n_1-1,\dots,n_P-1$; and $c$ is a non-zero complex number. Here the parameters $a_j$, $b_l$, $c$, and $A_0$ depend on $t$ in general. It follows from the properties of elliptic functions that Set $A_k=f(a_k)$, $1\leqslant k\leqslant m$.Making a linear transformation of the $z$-plane we can achieve that
$$
\begin{equation*}
\omega_1=\omega_1(t)\equiv 1 \quad\text{and}\quad b_0=b_0(t)\equiv0.
\end{equation*}
\notag
$$
In addition, making a shift in the target plane we can achieve that $A_0=A_0(t)\equiv 0$. Now assume that we know the dependencies $A_j(t)$ as smooth functions of the parameter $t$. As in the simply connected case, let $\dot{A}_j$ denote their $t$-derivatives. Set
$$
\begin{equation*}
G(z):=\frac{\prod_{l=0}^P\sigma^{n_l}(z-b_l)} {\prod_{j=0}^N\sigma^{m_j}(z-a_j)}\quad\text{and}\quad {G}_k(z):=(z-a_k)^{m_k}G(z).
\end{equation*}
\notag
$$
For each $k$ the function ${G}_k$ has a removable singularity at $a_k$, so it extends to this point as an analytic function. Let $\zeta(z)=\zeta(z;\omega_1,\omega_2)$ denote the Weierstrass $\zeta$-function with periods $\omega_1$ and $\omega_2$. Let $\eta_k:=2\zeta(\omega_k/2)$, $k=1,2$, and Theorem 10.3 ([119]). The one-parameter family of functions $f(z,t)$ defined by (40) satisfies the equation Consider the functions
$$
\begin{equation*}
L_k(\xi)=Z(\xi,a_k), \quad \tilde{L}_k(\xi)=Z(\xi,a_k)+\frac{1}{\xi-a_k}\,,\quad\text{and} \quad J_l(\xi)=Z(\xi,b_l).
\end{equation*}
\notag
$$
Theorem 10.3 yields the following result. Theorem 10.4 ([119]). The parameters in the integral representation (40) satisfy the system of ordinary differential equations In (44) we denote by $\mathfrak{P}$ the Weierstrass $\mathfrak{P}$-function; ${\partial\log\sigma(a_j)}/{\partial\omega_2}$ is the derivatives of $\log\sigma(z)=\log\sigma(z;\omega_1,\omega_2)$ with respect to $\omega_2$; this derivative can be found by the formula (see [116], Theorem 3)
$$
\begin{equation*}
\frac{\partial \log\sigma(z)}{\partial\omega_2}= -\frac{1}{2\pi i}\biggl[\frac{1}{2}\,\omega_1\bigl(\mathfrak{P}(z)- \zeta^2(z)\bigr)+\eta_1(z\,\zeta(z)-1)-\frac{g_2}{24}\,\omega_1z^2\biggr],
\end{equation*}
\notag
$$
where $g_2$ is the Weierstrass invariant:
$$
\begin{equation}
{g_2}={60}\sum_{\omega\in\Omega\setminus\{0\}}\frac{1}{\omega^4}\,,
\end{equation}
\tag{45}
$$
where the sum is taken over all non-zero points of the lattice $\Omega$ generated by $\omega_1$ and $\omega_2$. Among (41)–(44) we distinguish the second equality in (43), which describes the variation of an important characteristic of conformal tori, the so-called conformal modulus $m=\omega_2/\omega_1$, under the change of the critical values of an elliptic function of the form (40). Since we have set $\omega_1\equiv 1$, it follows that $\omega_2=m$ and
$$
\begin{equation}
\dot{m}=\frac{2\pi i}{c}\sum_{k=1}^N\frac{{G}^{(m_k-1)}_k(a_k)}{(m_k-1)!}\,.
\end{equation}
\tag{46}
$$
The family of functions mapping an annulus onto the exterior of two straight line cuts. Note that families of elliptic functions can be used not only as maps of tori, but also to map doubly connected domains. We consider conformal mappings $g$ of an annulus $\{q<|\zeta|<1\}$ onto the exterior $G=G(A_1,A_2,A_3,A_4)$ of two disjoint line segments $A_1A_2$ and $A_3A_4$ on the $w$-plane. Using the auxiliary map $z\mapsto \zeta=\exp\{2\pi i z\}$, in place of $g$ we consider the mapping $f:=g(2\pi i z)$ defined in the strip
$$
\begin{equation*}
S:=\{-m<\operatorname{Im} z<0\},\quad m=\frac{1}{2\pi}\,\log (q^{-1}).
\end{equation*}
\notag
$$
It takes the rectangle $\Pi=\{0<\operatorname{Re} z<1, \,-m<\operatorname{Im} z<0\}$ with identified vertical sides onto the domain $G$ (see Fig. 1). Let $\beta$ be the angle between the lines containing the segments. Note that $m$ is the conformal modulus of the doubly connected domain $G$. We can show that $f$ has the form
$$
\begin{equation}
f(z)= C\int_{0}^z e^{\gamma\xi}\,\frac{\prod_{k=1}^4\sigma(\xi-z_k)} {\sigma^2(\xi-z_0)\sigma^2(\xi-\overline{z}_0)}\,d\xi+C_1.
\end{equation}
\tag{47}
$$
Here $\sigma(z)=\sigma(z;1,2im)$ is the Weierstrass $\sigma$-function with periods $\omega_1=1$ and $\omega_2=2im$. Moreover, $\gamma=\beta\eta_1/\pi$, where $\eta_1=2\zeta(\omega_1/2;1,2im)$ and $\zeta(z;1,2im)$ is the Weierstrass $\zeta$-function with periods $\omega_1=1$ and $\omega_2=2im$. The points $z_k=x_k+i y_k$ correspond to the endpoints $A_k$ of the line segments and satisfy the relations
$$
\begin{equation*}
z_1=x_1,\quad z_2=x_2,\quad z_3=x_3+i m,\quad\text{and}\quad z_4=x_4-i m
\end{equation*}
\notag
$$
for some real $x_k$ such that
$$
\begin{equation*}
\sum_{k=1}^4x_k=\frac{\beta}{\pi}\,,\quad m=\frac{1}{2\pi}\log(q^{-1});
\end{equation*}
\notag
$$
the point $z_0=i y_0$ is the unique pole; $C\ne 0$ and $C_1$ are complex constants. We can assume without loss of generality that $x_0=\operatorname{Re} z_0=0$. Now consider the one-parameter family of maps of the form (47) described above and satisfying the following conditions: the points $A_k=A_k(t)$ are smooth functions of the parameter $t$, for variable $t$ the straight lines containing the segments $A_1A_2$ and $A_3A_4$ are fixed. The following theorem was established in [48]. Theorem 10.5. The one-parameter family of functions $f(z,t)$ satisfies the partial differential equation where Relying on Theorem 10.5, we can find a family of ordinary differential equations for the parameters $x_k$, $1\leqslant k\leqslant 4$, $c$, and $m$; it is too bulky to present it here, so we refer the reader to [48]. Using this system one can effectively find approximate values of the parameters in (47) and, in particular, the conformal modulus of the domain $G(A_1,A_2,A_3,A_4)$ in question. Note that relation (48) yields in fact a formula for the variation of the conformal modulus of the doubly connected domain $G=G(A_1,\kern-1pt A_2,\kern-1pt A_3,\kern-1pt A_4)$ under smooth variation of the endpoints $A_j$ of the segments; it can be used in investigations of the monotonicity of the conformal modulus of domains of the form $G(A_1,\kern-1pt A_2,\kern-1pt A_3,\kern-1pt A_4)$. Consider the following problem. Let $A_3A_4$ be a fixed line segment lying fully in the right-hand half-plane and intersecting the real axis at a point $\widetilde{x}$ so that one of its endpoints lies on the imaginary axis. Let $A_1A_2$ be an interval $[a- l/2,a+ l/2]$ of the real axis of fixed length $l$. How does the conformal modulus of the domain $G(A_1,A_2,A_3,A_4)$ change under the change of the parameter $a$? On the basis of the variational formula (48), using the technique of polarization (see [56], Theorem 1.2) we can obtain the following result. Theorem 10.6 ([48]). If $\widetilde{x}\leqslant l/2$, then for $a$ varying from $-\infty$ to $\widetilde{x}-l/2$ the conformal modulus of the domain $G(A_1,A_2,A_3,A_4)$ decreases from $+\infty$ to $0$. On the other hand, if $\widetilde{x}> l/2$, then the conformal modulus decreases from $+\infty$ to a positive quantity as $a$ varies from $-\infty$ to $0$. 11. Critical values of polynomialsIn 1981 Smale [140] stated the following famous conjecture on critical values of polynomials. Consider polynomials $f$ of degree $n \geqslant 2$ such that $f(0)=0$ and $f'(0)=1$. Set
$$
\begin{equation*}
S(f)=\min\biggl\{\biggl|\frac{f(\zeta)}{\zeta}\biggr| \colon f'(\zeta)=0 \biggr\}
\end{equation*}
\notag
$$
and Smale’s conjecture claims that $K_n=1-1/n$. Using methods of geometric function theory and Bieberbach’s inequality for the second coefficient in the class $S$, Smale showed that for polynomials $f$ we have $S(f)<4$. The lower bound for $K_n$ is attained at the polynomials $f(z)=z+cz^n$ where $c>0$ is arbitrary; for them $S(f)=1-1/n$. Thus, we have the bounds As concerns refining the upper bounds for $K_n$, there are many results in this direction, but the constant $4$ has not been improved asymptotically yet. In addition, there are many results on certain subclasses of polynomials. Sikorav (see [145]) showed that if $n \leqslant 4$, then $K_n=1-1/n$ and, moreover, except for $f(z)=z+c z^n$, at some critical point $\zeta$ of $f$ we always have the strict inequality
$$
\begin{equation*}
\biggl|\frac{f(\zeta)}{\zeta}\biggr| < 1-\frac{1}{n}\,.
\end{equation*}
\notag
$$
Numerical experiments carried out by Marinov and Sendov [137] showed that, with high probability, Smale’s conjecture holds for all $n\leqslant 10$. Beardon, Minda, and Ng [30] showed that $K_n\leqslant 4^{1-1/n}$; this inequality was slightly improved in [41] and [71]. The best upper bounds were obtained in [42] and [43], but they do not improve over the constant 4 for large values of $n$ either. If all the critical values of $f$ have equal moduli, or all critical points lie on the same circle, then a theorem of Sheil-Small ([138], pp. 361–362) claims that $S(f)< 1$. Using symmetrization methods, which is quite a non-standard approach in this field, Dubinin [55] solved Smale’s extremal problem in a slightly different setting. It follows directly from his result that $S(f)\leqslant 1-1/n$ in the case when the critical values or critical points have the same modulus. Tischler [145] proved that $S(f)< 1$ in the case when all the zeros of the polynomial $f$ lie on one circle. In the case when $f$ has only real zeros Palais (see [139], p. 159) proved that $S(f)<1$. In the same case Tischler [145] obtained a slightly better estimate $S(f)\leqslant 1-1/n$. Note that by Rolle’s classical theorem, if all the zeros of the polynomial $f$ are real, then its critical points are too. Since the converse result fails in general, the case of real critical points is more general than the case of real zeros of a polynomial. Under the assumption that all critical points of $f$ are real, Sheil-Small ([138], p. 368) showed that $S(f)< e-2$, after which Rahman and Schmeisser ([135], p. 217) established a slightly better estimate:
$$
\begin{equation*}
S(f) \leqslant \frac{n-2}{n} \biggl[\biggl(\frac{n-1}{n-2}\biggr)^{n-1}-2\biggr] < e-2,
\end{equation*}
\notag
$$
where $n\geqslant 3$. The definitive result in the case of real critical points is due to Hinkkanen and Kayumov [80]. The following theorem holds. Theorem 11.1. Let $f$ be a polynomial of degree $n \geqslant 3$ normalized so that $f(0)=0$ and $f'(0)=1$. Assume that all of its critical points are real. Then there is a critical point $\zeta$ such that except for $f(z)=z-c z^3$, when equality holds for each $c>0$. Now assume that the critical points of $f$, that is, the zeros of $f'$, lie in the union of $k$ rays going out of the origin. It was conjectured in [80] than then $S(f)\leqslant 1-1/(k+1)$, so that $K_n=1-1/n$. In that paper this conjecture was proved for $k=1$ and $2$. In Smale’s original question the above normalization is not assumed. Now let $f$ be an arbitrary polynomial of degree $n \geqslant 2$. Take any point $t$ in the complex plane $\mathbb{C}$ such that $f'(t)\ne 0$. Consider
$$
\begin{equation*}
S(f,t)=\min\biggl\{\biggl|\frac{f(\zeta)-f(t)}{(\zeta-t)f'(t)}\biggr|\colon f'(\zeta)=0\biggr\}.
\end{equation*}
\notag
$$
Since we can replace $f$ by $af+b$ and the variable $z$ by $cz+d$ for any complex numbers $a$, $b$, $c$, and $d$ such that $ac\ne 0$, it is easy to see that $S(f,t)=S(g,0)$ for a polynomial $g$ of degree $n$ such that $g(0)=0$ and $g'(0)=1$. Now we can set $g(z)=(f(\beta z+t)-f(t))/(\beta f'(t))$ for any $\beta\in \mathbb{C}\setminus \{0\}$. However, the positions of critical points of $g$ are different from those of critical points of $f$, so if we take critical points of $f$ on the union of $k$ rays issuing from $0$ to infinity, then the critical points of $g$ lie in the union of $k$ rays from some point $\alpha$, which does not coincide with $0$ in general. Note that if $\alpha\ne 0$, then making an appropriate choice of $\beta$ we can assume that $\alpha=1$. We consider only the case when $k=1$ or $k=2$. If $\alpha=0$, then going over to $g$ we see that this problem was already solved in [80]. So we consider the case $\alpha\ne 0$. Let $k=1$, so that there is a unique ray, lying on some straight line, or $k=2$, when two rays form a straight line. If this line passes through the origin, then we are again in the case described in [80]. So assume that this line does not pass through the origin. Then we can replace $\alpha$ by the point on this line which is the closest one to the origin. Making a further rotation and a dilation we can assume that $\alpha$ is equal to $1$ and the line is vertical. In [82] the problem of estimating $S(g,0)$ was considered in the case when the critical points of $g$ lie on the union of one or two rays issuing from some point and the problem when the critical points lie in some sector. Now assume again that the polynomial $f$ is normalized so that $f(0)=0$ and $f'(0)=1$. If $n=5$ and the critical points of $f$ are $\pm 1$ and $\pm i$, then $S(f)=4/5$. Shifting slightly the critical point $z=1$ and forming a new normalized polynomial $f_1$ we can still make $S(f)$ arbitrarily close to $4/5$. Then the straight line $L_1$ containing $-1$ and $i$ and the straight line $L_2$ containing $-i$ and the new critical point, which is close to $1$ but does not lie on the line connecting $-i$ and $1$, intersect at some point $\alpha$. Thus, the critical points of $f_1$ lie in the union of two rays issuing from $\alpha$ and going to infinity. This shows that for $k=2$ the best constant is at least $4/5$. For $k=2$ we can obtain better estimates by imposing additional conditions on the configuration of two rays. The following results hold [81]. Theorem 11.2. Let $f$ be a polynomial of degree $n \geqslant 2$ such that $f(0)=0$ and $f'(0)=1$. Assume that the critical points of $f$ lie in the sector $\{re^{i\theta}\colon r> 0, |\theta|\leqslant \pi/6\}$. Then $S(f)\leqslant 1/2$ and equality holds if and only if $n=2$. Theorem 11.3. Let $f$ be a polynomial of degree $n \geqslant 2$ such that $f(0)=0$ and $f'(0)=1$. Assume that the critical points of $f$ lie on the ray $\{1+re^{i\theta}\colon r\geqslant 0\}$, where $0\leqslant \theta\leqslant \pi/2$. Then $S(f)\leqslant 1/2$ and equality holds if and only if $n=2$. Theorem 11.4. Let $f$ be a polynomial of degree $n \geqslant 2$ such that $f(0)=0$ and $f'(0)=1$. Assume that the critical points of $f$ lie on the union of the rays $\{1+re^{\pm i\theta}\colon r\geqslant 0\}$, where $0<\theta\leqslant \pi/2 $. Then $S(f) < 2/3$. Of course, if $0\leqslant \theta \leqslant \pi/6$ in the last theorem, then we are in the case covered by Theorem 11.2. In this case we have the upper bound $1/2$ rather than $2/3$. Now we describe results related to the so-called dual Smale conjecture. In fact, there is a certain analytic relation for critical values of a polynomial (a rather complicated one, which can only be written explicitly for small degrees of the polynomial, for instance, for degree at most 4). As we have already seen, this relation does not allow all quantities $|f(\zeta)/\zeta|$ (where $\zeta$ is a critical point of the polynomial $f$) to be arbitrarily large simultaneously. For the same reason all these quantities cannot be too small simultaneously. This was noticed by Tischler in the case of so-called conservative polynomials, that is, polynomials such that at each critical point, where $C$ is some constant. Tischler [145] showed that in this case $C \geqslant 1/n$.Dubinin and Sugawa [61] and, independently, Ng put forward the following conjecture. Conjecture 11.1. Let $f$ be a polynomial of degree $\deg f\geqslant2$ such that $f(0)=0$ and $f'(0)=1$. Then $f$ has a critical point $\zeta$ such that The same authors found a lower bound: they showed that there exists a critical point $\zeta$ such that
$$
\begin{equation*}
\biggl|\frac{f(\zeta)}{\zeta}\biggr|\geqslant\frac{1}{n\,4^n}\,;
\end{equation*}
\notag
$$
this inequality was slightly improved in [121]. To date, the best estimate in this conjecture is due to Dubinin [58]: he proved that there exists a critical point $\zeta$ such that
$$
\begin{equation*}
\biggl|\frac{f(\zeta)}{\zeta}\biggr|\geqslant \frac{1}{n}\tan\frac{\pi}{4n}\,.
\end{equation*}
\notag
$$
This improves significantly the previous results obtained in this direction, but it is still quite far from the conjectural inequaliy, even in the sense of order. Now we present precise results in this area. For small values of $n$ Conjecture 11.1 is easy to verify directly. However, starting from $n=4$ it becomes more difficult, and more complicated calculations are required. For $n=4$ the result follows from an inequality in [145]. Namely, it was shown there that there exists a critical point $\zeta$ at which
$$
\begin{equation*}
\biggl|\frac{f(\zeta)}{\zeta}-\frac{1}{2}\biggr| \leqslant \frac{1}{2}-\frac{1}{n}\,, \qquad 2 \leqslant n \leqslant 4.
\end{equation*}
\notag
$$
Unfortunately, Tyson showed that this result fails for $n \geqslant 5$. It turns out that for $n \leqslant 7$ this problem reduces to an extremal problem of classical type, namely, we can present a deterministic algorithm for the choice of a critical point, which is based on choosing a critical point with the least modulus. Thus the problem reduces to a multiparameter extremal problem in the disc. Kayumov, Khammatova, and Hinkkanen [82] established the following result. Theorem 11.5. Let $f$ be a polynomial of degree six such that $f(0)=0$ and $f'(0)= 1$. Then there exists a point $\zeta$ such that $f'(\zeta)=0$ and $|f(\zeta)/\zeta| \geqslant 1/6$. Moreover, a critical point $\zeta$ such that $|f(\zeta)/\zeta | > 1/6$ exists, provided $f$ has a form distinct from where $a\in {\mathbb C}\setminus \{0\}$. The case $n=5$ is similar but easier to treat than $n=6$. The case $n=7$ has only been dealt with numerically; no published analytic proof is available. 
Citation in format AMSBIB
\Bibitem{AvkKayNas23} |
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