A New Method for Estimating the Coefficients of Holomorphic Functions

Abstract

The paper provides a new approach to estimating the coefficients of arbitrary holomorphic functions, which still remains an important problem of complex analysis. This approach is intrinsically connected with the features of univalent functions and with Teichmüller spaces.

1. Introductory Remarks

Estimating coefficients of holomorphic functions is an old problem with a two hundred years history, but it still remains important for many geometric and physical applications of complex analysis, for example, in view of their connection with string theory and with a holomorphic extension of Virasoro algebra. The Taylor coefficients reflect the fundamental intrinsic features of holomorphic maps and conformality and are still investigated from different points of view (see, e.g., [1,2,3,4,5,6,7,8,9,10,11]).

We provide a new approach to estimating the coefficients on rather general classes of holomorphic functions using the deep features of univalent functions and Teichmüller space theory.

We start with holomorphic functions

$$f\left(z\right)=c_0+c_{1}z+\dots +c_n{z}^n+\dots$$

in the unit disk $\mathbb{D}=\left\{\right|z|<1\}$ having the uniform growth

$${\displaystyle f\left(z\right)={O(1/(1-\left|z\right|}^2\left)\right))\mathrm{as}|z|\to 1.}$$

These functions are hyperbolically bounded and form the complex Banach space ${\mathbf{B}}_2$ with norm

$${\parallel f\parallel }_{{\mathbf{B}}_2}=\underset{\mathbb{D}}{\sup }{(1-|z|}^2{)}^2\left|f\left(z\right)\right|,$$

which is dual to the subspace $A_1\left(\mathbb{D}\right)$ of integrable holomorphic functions (more precisely, of quadratic differentials $f\left(z\right)d{z}^2$) in $L_1\left(\mathbb{D}\right)$.

The mean value inequality for holomorphic functions yields that all f from $A_1\left(\mathbb{D}\right)$ have the growth (1); hence, $A_1\left(\mathbb{D}\right)\subset {\mathbf{B}}_2$; see, e.g., [12].

The space ${\mathbf{B}}_2$ has a special interest because its points are the Schwarzian derivatives

$$S_w\left(z\right)={\left({\displaystyle \frac{w^{″}\left(z\right)}{w^{″}\left(z\right)}}\right)}^{\prime }-{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{w^{″}\left(z\right)}{w^{\prime }\left(z\right)}}\right)}^2$$

of locally univalent holomorphic functions $w\left(z\right)=a_0+a_{1}z+\dots$ on $\mathbb{D}$, which are determined by $f\in {\mathbf{B}}_2$ as solutions of the differential equation

$$(w^{″}\left(z\right)/w^{\prime }{\left(z\right))}^{\prime }-(w^{″}\left(z\right)/w^{\prime }{\left(z\right))}^2/2=f\left(z\right),$$

(1)

satisfying the prescribed initial conditions

$$w\left(0\right)=a_0,w^{\prime }\left(0\right)=a_1,w^{″}\left(0\right)=a_2.$$

(2)

Equivalently, w is the ratio ${\eta }_1/{\eta }_2$ of two linearly independent solutions ${\eta }_1,{\eta }_2$ of the linear differential equation

$$2{\eta }^{″}\left(z\right)+f\left(z\right)\eta \left(z\right)=0$$

(3)

with ${\eta }_1\left(0\right)=0,{\eta }_1^{\prime }\left(0\right)=1$ and ${\eta }_1\left(0\right)=1,{\eta }_1^{\prime }\left(0\right)=0$.

The coefficients $c_n$ are uniquely determined by coefficients $a_m$ of the corresponding $w\left(z\right)$, and the problem is to find the explicit bounds for $c_n$ having the bounds for $a_m$.

We shall also use the inverted functions

$$W_w\left(z\right)=1/w(1/z)=b_0+b_1{z}^{-1}+\dots +b_n{z}^{-n}+\dots ,\left|z\right|>1,$$

of functions $w\left(z\right)\in S$, which are nonvanishing (zero free) and univalent on the complementary disk ${\mathbb{D}}^{*}=\{z\in \widehat{\mathbb{C}}=\mathbb{C}\cup \{\infty \}:\left|z\right|>1\}$.

The class of all $\widehat{\mathbb{C}}$-holomorphic univalent functions on ${\mathbb{D}}^{*}$ with a simple pole at infinity is denoted by $\mathrm{\Sigma }$. Let $S_Q$ and ${\mathrm{\Sigma }}_Q$ denote the (dense) subclasses from S and $\mathrm{\Sigma }$ formed by functions admitting quasiconformal extension to the whole Riemann sphere $\widehat{\mathbb{C}}$.

Then, we have the unit balls $\mathrm{Belt}{\left({\mathbb{D}}^{*}\right)}_1$ and $\mathrm{Belt}{\left(\mathbb{D}\right)}_1$ of Beltrami coefficients $\mu \in L_{\infty }\left(\mathbb{C}\right)$ supported on the disks ${\mathbb{D}}^{*}$ and $\mathbb{D}$, and the indicated functions are the restrictions of solutions $w^{\mu }$ of the Beltrami equation ${\partial }_{\overline{z}}w=\mu {\partial }_{z}w$ (conformal on $\mathbb{D}$ and ${\mathbb{D}}^{*}$, respectively).

Quasiconformal maps $w^{\mu }\left(z\right)$ defined by $\mu \in \mathrm{Belt}{\left({\mathbb{D}}^{*}\right)}_1$ are the generalized homeomorphic solutions of the indicated Beltrami equation on the complex plane $\mathbb{C}$, and the partial derivatives ${\partial }_{z}w,{\partial }_{\overline{z}}w$ are understood in the distributional sense. So, w is determined by $\mu$ up to a Moebius (fractional linear) transformation of $\widehat{\mathbb{C}}$.

The Schwarzians $S_w$ of $w\in S_Q$ run over a bounded domain in ${\mathbf{B}}_2$ modeling the universal Teichmüller space $\mathbf{T}$ (and similarly $S_W$ of $W\in {\mathrm{\Sigma }}_Q$). The Schwarzian Equation (1) also determines w up to a Moebius transformation.

2. Main Distortion Theorems and Beyond

Special Subclasses of ${\mathbf{B}}_2$

Now, consider the spaces ${\mathbf{B}}_q$ of holomorphic functions $f\left(z\right)$ on $\mathbb{D}$ with the smaller growths

$${\displaystyle f\left(z\right)={O(1/(1-\left|z\right|}^p\left)\right)\mathrm{as}\left|z\right|\to 1}$$

(4)

with $0\le p<2$.

Our aim is to prove that for such spaces the connection between the extremal univalent functions maximizing $a_2$ and the Schwarzians with extremal $c_1$ is intrinsic. For the technical reasons, we need to distinguish some subclasses of ${\mathbf{B}}_p$ formed by the integrable functions.

Namely, take $q<p$, and let ${\mathbf{B}}_{p,q}$ be the (linear) subspace of ${\mathbf{B}}_p$ formed by functions $f\in {\mathbf{B}}_p$ with finite $L_q$ norm

$${\parallel f\parallel }_{p,q}={\left(\underset{\mathbb{D}}{\int \int }{\left|f\left(z\right)\right|}^{q}dxdy\right)}^{1/q}.$$

For such classes, the indicated connection is given by the following:

Theorem 1.

(a) For any connected submanifold G in the space ${\mathbf{B}}_{p,q}$ with even $q=2s\ge 2(s\in \mathbb{N})$, whose boundary has the common points with $\partial \mathbf{T}$, the function $f_0\left(z\right)=c_0+c_{1}z+\dots \in \overline{G}$ maximizing the first coefficient $c_1$ on $\overline{G}$ is the Schwarzian $S_{w_0}$ of a univalent function $w_0\left(z\right)$ on $\mathbb{D}$ with maximal second coefficient $a_2$ among all univalent functions w with the Schwarzians $S_w\in \overline{G}$ (here $\overline{G}$ is the closure of G in ${\mathbf{B}}_{p,q}$-norm).(b) The coefficients $a_m$ and non-zero $c_n$ are estimated for all $m\ge 3$ and $n\ge 2$ by

$$|a_m|\le |a_m^0|$$

(5)

and

$$|c_n|\le \max (|c_1^0|,|c_n^0\left|\right).$$

(6)

(c) If f is zero-free (does not vanish) on $\mathbb{D}$, then the assertion $\left(a\right)$ and the estimates (5), (6) are valid for all $q\ge 2$.

In particular, if $|c_n^0|\le |c_1^0|$ for all $n\ge 2$, then all $c_n$ are bounded by this $c_1^0$.

This theorem provides a way of solving the indicated problem of estimating the coefficients $c_n$ (for spaces with $L_q$-norm), reducing this problem to establishing the extremal $f_0$ only for the first coefficient $c_1$. Then, it remains to solve Equations (1) or (2) for this $f_0$.

Another goal is to use the widely developed methods of geometric function theory.

Going to the limit as $q\to \infty$, one obtains, taking into account the compactness of ${\mathbf{B}}_p$ in the topology of locally uniform convergence on $\mathbb{D}$, the coefficient bounds for the whole initial class ${\mathbf{B}}_p$.

Theorem 2.

The assertions $\left(a\right),\left(b\right),\left(c\right)$ of Theorem 1 are valid for all functions $f\in {\mathbf{B}}_p$ with $p<2$.

Note that the relation $c_1\left(S_{w_0}\right)\ne 0$ underlying the proof of Theorem 1 is not preserved under the limit as $q\to \infty$ and for collections of holomorphic functions with a growth greater than (4). For example, the Koebe function

$${\kappa }_{\theta }\left(z\right)={\displaystyle \frac{z}{{(1-e^{i\theta }z)}^2}}=z+{\displaystyle \sum _2^{\infty }}m{e}^{-i(m-1)\theta }{z}^m,-\pi <\theta \le \pi ,$$

mapping the unit disk onto the complement of the ray $w=-t{e}^{-i\theta },{\displaystyle \frac{1}{4}}\le t\le \infty$, is extremal for many holomorphic functionals $J\left(f\right)$ on the class S; in particular, it maximizes the coefficients $a_n,n>1$. But its Schwarzian

$$S_{{\kappa }_{\theta }}\left(z\right)=-{\displaystyle \frac{6e^{-2i\theta }}{{(1-e^{2i\theta }{z}^2)}^2}}=-6{\displaystyle \sum _0^{\infty }}(m+1)e^{i2m\theta }{z}^{2m}$$

is an even function (hence, $c_1=0$). The even coefficients of the Schwarzians $S_w$ on S are maximized by this function.

3. Geometric Features of Functions from ${\mathbf{B}}_p$

Recall that, by assumption, $0\le p<2$. For such p, the functions $f\in {\mathbf{B}}_p$ as elements of universal space $\mathbf{T}$ determine the univalent solutions of (1) or (3) having some remarkable properties (causing, among other things, that the union ${\bigcup }_p{B}_p$ is not dense in ${\mathbf{B}}_2$).

Namely, these solutions $w\left(z\right)$ are asymptotically conformal on the boundary, i.e., map the unit circle ${\mathbb{S}}^1$ onto the asymptotically conformal Jordan curves L, which means that, for any pair of points $a,b\in L$, we have

$$\underset{z\in L(a,b)}{\max }{\displaystyle \frac{|a-z|+|z-b|}{|a-b|}}\to 1\mathrm{as}|a-b|\to 0,$$

where the point z lies on L between a and b.

Such curves are quasiconformal circles without corners and can be rather pathological (see, e.g., [13], p. 249). All $C^1$-smooth curves are asymptotically conformal. Asymptotical conformality provide some specific features of univalent functions w and curves $w({\mathbb{S}}^1$, first of all,

$$S_w\left(z\right)\to 0\mathrm{as}\left|z\right|\to 1\mathrm{in}{\mathbf{B}}_2\mathrm{norm}.$$

If $\parallel S_w{\parallel }_{{\mathbf{B}}_2}<2$, then w has the Ahlfors–Weill quasiconformal extension $\widehat{w}$ to ${\mathbb{D}}^{*}$ with Beltrami coefficient

$${\mu }_w\left(z\right)=-{\displaystyle \frac{1}{2}}{\left(\right|z|}^2{-1)}^2\phi (1/\overline{z})(1/{\overline{z}}^4),z\in {\mathbb{D}}^{*};$$

see [14] (the Beltrami coefficients of such form are called harmonic).

4. Applications to Nonvanishg Holomorphic Functions

Such functions arise and play an essential role in many fields of complex analysis (in particular, as the derivatives of locally univalent functions).

The problem stated above includes the eminent Krzyz and Hummel–Scheinberg–Zalcman conjectures for nonvanishing holomorphic functions from the Hardy spaces $H^p$. The norm in these spaces is defined by

$${\parallel f\parallel }_p=\underset{r<1}{\sup }{\left({\displaystyle \frac{1}{2\pi }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}}{\left|f\left(r{e}^{i\theta }\right)\right|}^{p}d\theta \right)}^{1/p}$$

for $p<\infty$ and ${\parallel f\parallel }_{\infty }$ coincides with $L_{\infty }$-norm of this function.

The Krzyz conjecture [8] of 1968 states that Taylor’s coefficients of nonvanishing holomorphic functions $f\left(z\right)={\displaystyle \sum _0^{\infty }}{c}_n{z}^n\in H^{\infty }$ with ${\parallel f\parallel }_{\infty }\le 1$ are sharply estimated by

$$|c_n|\le 2/e,$$

(7)

with equality only for the function ${\kappa }_{\infty }\left(z^n\right)$, where

$${\kappa }_{\infty }\left(z\right)=\exp \left({\displaystyle \frac{z-1}{z+1}}\right)={\displaystyle \frac{1}{e}}+{\displaystyle \frac{2}{e}}z-{\displaystyle \frac{2}{3e}}{z}^3+...$$

(and its compositions with rotations about the origin).

Its deep generalization to spaces $H^p$ with $p>1$ is given by the Hummel–Scheinberg–Zalcman conjecture posed in 1977, which states for all nonvanishing $f\in H^p$ with ${\parallel f\parallel }_p\le 1$ the sharp estimate

$$|c_n{|\le (2/e)}^{1-1/p},$$

and this bound is realized only by the functions ${ϵ}_2{\kappa }_{n,p}\left({ϵ}_{1}z\right)$, where $|{ϵ}_1|=|{ϵ}_2|=1$ and

$${\kappa }_{n,p}\left(z\right)={\left[{\displaystyle \frac{{(1+z^n)}^2}{2}}\right]}^{1/p}{\left[\exp {\displaystyle \frac{z^n-1}{z^n+1}}\right]}^{1-1/p}.$$

Both conjectures have been investigated by many authors; the results have been established only for the initial coefficients $c_n$ (see, e.g., [6,10,15,16,17,18,19,20,21,22,23] and the references cited there) and have been proven by the author for $p\ge 2$ in [6,24] using Brown’s estimate of $c_1$ (for such p we have the situation described after Theorem 1).

Note also that the bound (7) is valid for a more general collection of nonvanishing functions containing also the unbounded functions which belong to all classes $H^p,p>1$.

We mention the following consequence from Theorem 1 for nonvanishing functions giving a weaker answer to the Hummel–Scheinberg–Zalcman conjecture for the remaining case $1<p<2$.

Corollary 1.

The estimate (6) is valid for nonvanishing functions $f\left(z\right)$ from the unit ball $B_1\left(H^p\right)$ of $H^p$ for any $p>1$.

Due to [16], $|c_1{|\le (2/e)}^{1-1/p}$, with equality only for the rotations of function ${\kappa }_{1,p}\left(z\right)$.

In the case $p\ge 2$, the boundary functions $f\left(e^{i\theta }\right)={\lim }_{r\to 1}f\left(r{e}^{i\theta }\right)$ of $f\in H^p$ admit Parseval’s equality

$$1\ge {\displaystyle \frac{1}{2\pi }}{\displaystyle \underset{-\pi }{\overset{\pi }{\int }}}|f\left(e^{i\theta }\right){|}^{2}d\theta ={\displaystyle \sum _0^{\infty }}{\left|c_n\right|}^2.$$

The expression of ${\kappa }_{1,p}\left(z\right)={\displaystyle \sum _0^{\infty }}{c}_n^0{z}^n$ yields

$$|c_1^0{|}^2={(2/e)}^{2(1-1/p)}=0.5041..{.}^{1-1/p}>0.5041\dots$$

(for all $p>1$); hence, ${\sum }_2^{\infty }|c_n^0{|}^2<0.5<{\left|c_1^0\right|}^2$, and all $c_n$ are estimated by $|c_1^0|$.

5. Digression to Teichmüller Spaces

We briefly recall the results from Teichmüller space theory involved in the proof of Theorem 1; the details can be found, for example, in [25,26,27]. It is technically more convenient to deal with functions from ${\mathrm{\Sigma }}_Q$.

The universal Teichmüller space $\mathbf{T}=\mathrm{Teich}\left(\mathbb{D}\right)$ is the space of quasisymmetric homeomorphisms of the unit circle ${\mathbb{S}}^1$ factorized by Möbius maps; all Teichmüller spaces have their biholomorphic copies in $\mathbf{T}$.

The canonical complex Banach structure on $\mathbf{T}$ is defined by factorization of the ball of the Beltrami coefficients (or complex dilatations)

$$\mathrm{Belt}{\left(\mathbb{D}\right)}_1=\{\mu \in L_{\infty }\left(\mathbb{C}\right):\mu |{\mathbb{D}}^{*}=0,\parallel \mu \parallel <1\},$$

letting ${\mu }_1,{\mu }_2\in \mathrm{Belt}{\left(\mathbb{D}\right)}_1$ be equivalent if the corresponding quasiconformal maps $w^{{\mu }_1},w^{{\mu }_2}$ (solutions to the Beltrami equation ${\partial }_{\overline{z}}w=\mu {\partial }_{z}w$ with $\mu ={\mu }_1,{\mu }_2$) coincide on the unit circle ${\mathbb{S}}^1=\partial {\mathbb{D}}^{*}$ (hence, on $\overline{{\mathbb{D}}^{*}}$). Such $\mu$ and the corresponding maps $w^{\mu }$ are called $\mathbf{T}$-equivalent.

The following important lemma from [24] allows one to use some other normalizations of quasiconformally extendable functions.

Lemma 1.

For any Beltrami coefficient $\mu \in \mathrm{Belt}{\left({\mathbb{D}}^{*}\right)}_1$ and any ${\theta }_0\in [0,2\pi ]$, there exists a point $z_0=e^{i\alpha }$ located on ${\mathbb{S}}^1$ so that $|e^{i{\theta }_0}-e^{i\alpha }|<1$ and such that for any θ satisfying $|e^{i\theta }-e^{i\alpha }|<1$ the equation ${\partial }_{\overline{z}}w=\mu \left(z\right){\partial }_{z}w$ has a unique homeomorphic solution $w=w^{\mu }\left(z\right)$, which is holomorphic on the unit disk $\mathbb{D}$ and satisfies

$$w\left(0\right)=0,w^{\prime }\left(0\right)=e^{i\theta },w\left(z_0\right)=z_0.$$

(8)

Hence, $w^{\mu }\left(z\right)$ is conformal and does not have a pole in $\mathbb{D}$ (so $w^{\mu }\left(z_{*}\right)=\infty$ at some point $z_{*}$ with $|z_{*}|\ge 1$).

In particular, this lemma allows one to define the Teichmüller spaces using the quasiconformally extendible univalent functions $w\left(z\right)$ in the unit disk $\mathbb{D}$ normalizing these functions by

$$w\left(0\right)=0,w^{\prime }\left(0\right)=e^{i\theta },w\left(1\right)=1.$$

All such functions are holomorphic in the disk $\mathbb{D}$.

The proof of Theorem 1 also involves other Teichmüller spaces. The corresponding space ${\mathbf{T}}_1=\mathrm{Teich}\left({\mathbb{D}}_{*}\right)$ for the punctured disk ${\mathbb{D}}_{*}=\mathbb{D}\setminus \left\{0\right\}$ is formed by classes ${\left[\mu \right]}_{{\mathbf{T}}_1}$ of ${\mathbf{T}}_1$-equivalent Beltrami coefficients $\mu \in \mathrm{Belt}{\left(\mathbb{D}\right)}_1$ so that the corresponding quasiconformal automorphisms $w^{\mu }$ of the unit disk coincide on both boundary components (unit circle ${\mathbb{S}}^1$ and the puncture $z=0$) and are homotopic on $\mathbb{D}\setminus \left\{0\right\}$. This space can be endowed with a canonical complex structure of a complex Banach manifold and embedded into $\mathbf{T}$ using uniformization of ${\mathbb{D}}_{*}$ by a cyclic parabolic Fuchsian group acting discontinuously on $\mathbb{D}$ and ${\mathbb{D}}^{*}$. The functions $\mu \in L_{\infty }\left(\mathbb{D}\right)$ are lifted to $\mathbb{D}$ as the Beltrami measurable $(-1,1)$-forms $\tilde{\mu }d\overline{z}/dz$ in $\mathbb{D}$ with respect to $\mathrm{\Gamma }$, i.e., via $(\tilde{\mu }\circ \gamma )\overline{{\gamma }^{\prime }}/{\gamma }^{\prime }=\tilde{\mu },\gamma \in \mathrm{\Gamma }$, forming the Banach space $L_{\infty }(\mathbb{D},\mathrm{\Gamma })$; we extend these $\tilde{\mu }$ by zero to ${\mathbb{D}}^{*}$. Then, ${\mathbf{T}}_1$ is canonically isomorphic to the subspace $\mathbf{T}\left(\mathrm{\Gamma }\right)=\mathbf{T}\cap {\mathbf{B}}_2\left(\mathrm{\Gamma }\right)$, where ${\mathbf{B}}_2\left(\mathrm{\Gamma }\right)$ consists of elements $\phi \in {\mathbf{B}}_2$, satisfying $(\phi \circ \gamma ){\left({\gamma }^{\prime }\right)}^2=\phi$ in ${\mathbb{D}}^{*}$ for all $\gamma \mathrm{\Gamma }$.

Due to the Bers isomorphism theorem, the space ${\mathbf{T}}_1$ is biholomorphically isomorphic to the Bers fiber space

$$\mathrm{Fib}\left(\mathbf{T}\right)=\{({\varphi }_{\mathbf{T}}\left(\mu \right),z)\in \mathbf{T}\times \mathbb{C}:\mu \in \mathrm{Belt}{\left(\mathbb{D}\right)}_1,z\in w^{\mu }\left(\mathbb{D}\right)\}$$

over the universal Teichmüller space $\mathbf{T}$ with holomorphic projection $\pi (\psi ,z)=\psi$ (see [25]).

This fiber space is a bounded hyperbolic domain in ${\mathbf{B}}_2\times \mathbb{C}$ and represents the collection of domains $D_{\mu }=w^{\mu }\left(\mathbb{D}\right)$ as a holomorphic family over the space $\mathbf{T}$. For every $z\in \mathbb{D}$, its orbit $w^{\mu }\left(z\right)$ in ${\mathbf{T}}_1$ is a holomorphic curve over $\mathbf{T}$.

The indicated isomorphism between ${\mathbf{T}}_1$ and $\mathrm{Fib}\left(\mathbf{T}\right)$ is induced by the inclusion map $j:{\mathbb{D}}_{*}\hookrightarrow \mathbb{D}$ forgetting the puncture at the origin via

$$\mu \mapsto (S_{w^{{\mu }_1}},w^{{\mu }_1}\left(0\right))\mathrm{with}{\mu }_1=j_{*}\mu :=(\mu \circ j_0)\overline{j_0^{\prime }}/j_0^{\prime },$$

(9)

where $j_0$ is the lift of j to $\mathbb{D}$.

The Bers theorem is valid for Teichmüller spaces $\mathbf{T}(X_0\setminus \left\{x_0\right\})$ of all punctured hyperbolic Riemann surfaces $X_0\setminus \left\{x_0\right\}$; we use only its special case.

The spaces $\mathbf{T}$ and ${\mathbf{T}}_1$ can be weakly (in the topology generated by the spherical metric on $\widehat{\mathbb{C}}$) approximate by finite dimensional Teichmüller spaces $\mathbf{T}(0,n)$ of punctured spheres (Riemann surfaces of genus zero)

$$X_{\mathbf{z}}=\widehat{\mathbb{C}}\setminus \{0,1,z_1\dots ,z_{n-3},\infty \}$$

defined by ordered n-tuples $\mathbf{z}=(0,1,z_1,\dots ,z_{n-3},\infty ),n>4$ with distinct $z_j\in \mathbb{C}\setminus \{0,1\}$ (the details see, e.g., in [6]).

Another canonical model of $\mathbf{T}(0,n)$ is obtained again using the uniformization. This space is biholomorphic to a bounded domain in the complex Euclidean space ${\mathbb{C}}^{n-3}$.

Note also that all Teichmüller spaces are complete metric spaces with intrinsic Teichmüller metric defined by quasiconformal maps. By the Royden–Gardiner theorem, this metric equals the hyperbolic Kobayashi metric determined by the complex structure (see, e.g., [26,28,29]).

6. Preliminary Lemmas

The following lemma reveals the intrinsic connection between the covering and distortion features of holomorphic functions.

Let G be a domain in a complex Banach space $X=\left\{\mathbf{x}\right\}$ and $\chi$ be a holomorphic map from G into the universal Teichmüller space $\mathbf{T}=\mathrm{Teich}\left(D\right)$ with the base point D modeled as a bounded subdomain of ${\mathbf{B}}_2$. Assume that $\chi \left(G\right)$ is a (pathwise connected) submanifold of finite or infinite dimension in $\mathbf{T}\left(D\right)$, and put

$$a_{2,\theta }^0=|a_2(\succ \left(G\right))|=\sup \{|a_2|:S_w\in \chi \left(G\right)\},$$

Lemma 2

([6]). Let $w\left(z\right)$ be a holomorphic univalent solution of the Schwarz differential equation $S_w\left(z\right)=\chi \left(\mathbf{x}\right)$ on D satisfying $w\left(0\right)=0,w^{\prime }\left(0\right)=e^{i\theta }$ with the fixed $\theta \in [-\pi ,\pi ]$ and $\mathbf{x}\in G$ (hence $w\left(z\right)=e^{i\theta }z+{\sum }_2^{\infty }{a}_n{z}^n$). Assume that $a_{2,\theta }^0\ne 0$, and let $w_0\left(z\right)=e^{i\theta }z+a_2^0{z}^2+\dots$ be one of the maximizing functions for $a_{2,\theta }^0$. Then:

(a) For every indicated function $w\left(z\right)$, the image domain $w\left(D\right)$ covers entirely the disk $D_{1/\left(2\right|a_{2,\theta }^0\left|\right)}=\left\{\right|w|<1/\left(2\right|a_{2,\theta }^0\left|\right)\}$.

The radius value $1/\left(2\right|a_{2,\theta }^0\left|\right)$ is sharp for this collection of functions and fixed θ, and the circle $\left\{\right|w|=1/\left(2\right|a_{2,\theta }^0\left|\right)$ contains points not belonging to $w\left(\mathbb{D}\right)$ if and only if $|a_2|=|a_{2,\theta }^0|$ (i.e., when w is one of the maximizing functions).

(b) The inverted functions

$$W\left(\zeta \right)=1/w(1/\zeta )=e^{i\theta }\zeta -a_2^0+b_1{\zeta }^{-1}+b_2{\zeta }^{-2}+\dots$$

with $\zeta \in D^{-1}$ map domain $D^{-1}$ onto a domain whose boundary is entirely contained in the disk $\left\{\right|W+a_{2,\theta }^0|\le |a_{2,\theta }^0\left|\right\}$.

(c) Let D be the unit disk $\mathbb{D}$ and

$$\phi \left(z\right)=\chi \left(\mathbf{x}\right)=c_0+c_{1}z+c_2{z}^2+\dots ,\left|z\right|<1.$$

Let a point ${\mathbf{x}}_0\in G$ correspond to the function $w_0\left(z\right)$ maximizing the quantity $|a_{2,\theta }^0|$ on $[-\pi ,\pi ]$. If its Schwarzian $S_{w_0}=\chi \left({\mathbf{x}}_0\right)=c_0^0+c_1^0+\dots$ satisfies $c_1^0\ne 0$, then its modulus $|c_1^0|$ is maximal on this class, i.e.,

$$|c_1^0|=\sup \{|c_1\left(w_{\alpha ,\beta }\right)|:S_w\in \chi \left(G\right);\alpha ,\beta \in [-\pi ,\pi ]\}.$$

The proof of parts (a), (b) follows the classical lines of Koebe’s $1/4$ theorem (cf. [1]). The proof of assertion (c) is complicated.

We shall need the existence of special quasiconformal deformations of functions $f\in {Ł}_{q,\delta }$ preserving their $L_{q,\delta }$ norm and with some prescribed properties. A general result on such deformation was established in [5] for even q, having the origins in the local existence theorems for Riemann surfaces with finitely generated fundamental groups in [30].

Consider the functions $f\left(z\right)\in L_{2m,\delta }\cap L_{\infty }\left(\mathbb{D}\right)$, with

$$\underset{\mathbb{D}}{\sup }\left|f\left(z\right)\right|={M>\parallel f\parallel }_{L_{2m,\delta }}.$$

Let E be a ring domain bounded by a closed curve $L\subset \mathbb{D}$ containing inside the origin and by the unit circle $S^1=\partial \mathbb{D}$. Let, in addition,

$${\mathbf{d}}^0=(0,1,0,...,0)=:\left(d_k^0\right)\in {\mathbb{R}}^{n+1},$$

and $\left|\mathbf{x}\right|$ denote the Euclidean norm in ${\mathbb{R}}^l$.

Lemma 3.

For any holomorphic function $f\left(z\right)={\displaystyle \sum _{k=j}^{\infty }}{c}_k^0{z}^k\in L_{2m,\delta }\left(E\right)\cap L_{\infty }\left(E\right)$ (with $c_j^0\ne 0,0\le j<n$ and $m\in \mathbb{N}$), which is not a polynomial of degree $n_1\le n$, there exists a positive number ${\epsilon }_0$ such that for every point

$${\mathbf{d}}^{\prime }=(d_{j+1}^{\prime },\dots ,,d_n^{\prime })\in {\mathbb{C}}^{n-j}$$

and every $a\in \mathbb{R}$ satisfying the inequalities

$$|{\mathbf{d}}^{\prime }|\le \epsilon ,|a|\le \epsilon ,\epsilon <{\epsilon }_0,$$

there exists a quasiconformal automorphism h of the complex plane $\widehat{\mathbb{C}}$, which is conformal in the disk

$$D_0=\{w:|w-c_0^0|<\underset{\mathbb{D}}{\sup }|f_0\left(z\right)|+|c_0^0|+1\}$$

(hence also outside of E) and satisfies the conditions:

(i) $h^{\left(k\right)}\left(c_0^0\right)=k!d_k=k!(d_k^0+d_k^{\prime }),k=j+1,\dots ,n$ (i.e., $d_1=1+d_1^{\prime }$ and $d_k=d_k^{\prime }$ for $k\ge 2)$;

(ii) ${\parallel h\circ f\parallel }_{L_{2m,\delta }}^{2m}={\parallel f\parallel }_{L_{2m,\delta }}^{2m}+a$.

Moreover, the above assertions are extended to all zero-free functions $f\left(z\right)$ on the entire disk $\mathbb{D}$, which belongs to $L_{q,\delta }\left(E\right)\cap L_{\infty }\left(E\right),q>1$.

Such type lemmas were applied earlier in the case of non-weighted spaces $L_{2m}\left(E\right)$ (with norm $\parallel f\parallel ={\left({\int \int }_E{\left|f\right|}^{2m}dxdy\right)}^{1/2m}$); its proof is given in [5] and essentially involves the assumption $m=q/2\in \mathbb{N}$. The presence of weight $\lambda \left(\right|z\left|\right)$ does not change the arguments applied there.

Using the appropriately thin rings E adjacent to the unit circle, one derives that for any bounded in $\mathbb{D}$ function $f\in L_{2m,\delta }$ there exists a $O\left(\epsilon \right)$-quasiconformal automorphism h of $\widehat{\mathbb{C}}$ satisfying the conditions $\left(i\right)$ and

$${\parallel h\circ f\parallel }_{L_{2m,\delta }}^{2m}={\parallel f\parallel }_{L_{2m,\delta }}^{2m}+O\left(\epsilon \right).$$

The origins of such deformations go back to the local existence theorems in [30] for Riemann surfaces with finitely generated fundamental groups.

In the case, when $f\left(z\right)={\displaystyle \sum _0^{\infty }}{c}_n{z}^n\in L_{q,\delta }\left(E\right),q>1$, extends to a zero free holomorphic function on $\mathbb{D}$, one can use the function

$$f_{q/2}\left(z\right):=f{\left(z\right)}^{q/2}=e^{(p/2)\log f\left(z\right)},$$

with a fixed branch of the logarithmic function, which also is single valued, holomorphic and zero free in the unit disk $\mathbb{D}$; we take the principal branch. Explicitly,

$$f_{q/2}\left(z\right)=c_0^{q/2}\left(1+{\displaystyle \frac{q}{2}}{\displaystyle \frac{c_1}{c_0}}z+\dots \right)=c_0\left(f_{q/2}\right)+c_1\left(f_{q/2}\right)z+\dots .$$

This function belongs to the space $L_{2,\delta }\left(E\right)$, and one can apply to it all arguments used in [6] in the proof of Lemma 3 for $q=2m$. The correspondence $f\left(z\right)\leftrightarrow f_{q/2}\left(z\right)$ creates a biholomorphism (one-to-one and open map) between the neighborhoods of the origin in ${\mathbb{C}}^{n+1}$ filled by collections $\mathbf{d}\left(f\right)=(c_0,\dots ,c_n)$ and $\mathbf{d}\left(f_{q/2}\right)=(c_0\left(f_{q/2}\right),\dots ,c_n\left(f_{q/2}\right))$.

The following lemma provides that the set $B_1^0\left({\mathbf{B}}_{p,q}\right)$ of nonvanishing functions in the unit ball of each space ${\mathbf{B}}_{p,q}$ is open.

Lemma 4.

Each point $f\in B_1^0\left({\mathbf{B}}_{p,q}\right)$ has a neighborhood $U(f,ϵ)$ in ${\mathbf{B}}_{p,q}$, which entirely belongs to $B_1^0\left({\mathbf{B}}_{p,q}\right)$, i.e., contains only nonvanishing ${\mathbf{B}}_{p,q}$ functions on the disk $\mathbb{D}$. Take the maximal balls $U(f,ϵ)$ with such property. Then their union

$${\mathcal{U}}_{p,q}=\bigcup _{f\in B_{*}\left({\mathbf{B}}_{p,q}\right)}U(f,ϵ)$$

is an open path-wise connected set, hence a subdomain of $B_1\left({\mathbf{B}}_{p,q}\right)$.

Such a lemma is proved in [24] for nonvanishing functions from spaces $H^q$. Its arguments are extended to weighted spaces ${\mathbf{B}}_{p,q}$ almost straightforwardly.

Proof of Theorem 1.

The proof follows the lines of [5,6] and involves holomorphic lifting the given functionals

$$J\left(f\right)=a_m,I\left(f\right)=c_n,\mathrm{with}m\ge 3,n>1$$

(10)

to Teichmüller spaces $\mathbf{T}$ and ${\mathbf{T}}_1=\mathrm{Fib}\left(\mathbf{T}\right)$. It consists of five steps. First we prove the parts $\left(a\right)$ and $\left(b\right)$.

Step 1: Lifting the functionals (10) to spaces $\mathbf{T}$ and $\mathrm{Fib}\left(\mathbf{T}\right)$.

We pass to a wider collection of quasiconformally extendable to $\widehat{\mathbb{C}}$ holomorphic univalent functions with expansions

$$w\left(z\right)=a_{1}z+a_2{z}^2+\dots ,z\in \mathbb{D},$$

with $|a_1|=1$ and $w\left(z_0\right)=z_0$ for some point $z_0\in {\mathbb{S}}^1$ (depending on w), completed in the topology of locally uniform convergence on $\mathbb{C}$. Denote this collection by $\widehat{S}\left(1\right)$. It is a disjunct union

$$\widehat{S}\left(1\right)=\bigcup _{-\pi \le \theta ,\alpha <\pi }{S}_{\theta ,\alpha },$$

where $S_{\theta ,\alpha }$ consists of univalent functions $w\left(z\right)=e^{i\theta }z+a_2{z}^2+\dots$ with quasiconformal extensions to $\widehat{\mathbb{C}}$ satisfying $w\left(1\right)=1$ and their rotations $w_{\alpha ,\alpha }\left(z\right)$.

This family $\widehat{S}\left(1\right)$ naturally relates to the initial class S, because every $w\in S$ has its representative $\widehat{w}$ in $\widehat{S}\left(1\right)$ (not necessarily unique) obtained by pre and post compositions of w with rotations $z\mapsto e^{i\alpha }z$ about the origin,

$$w\left(z\right)\mapsto w_{\tau ,\theta }\left(z\right)=e^{-i\theta }w\left(e^{i\tau }z\right)\mathrm{with}\tau =arg{z}_0,$$

(11)

where $z_0$ is a point for which $w\left(z_0\right)=e^{i\theta }$ is a common point of the unit circle and the boundary of domain $w\left(\mathbb{D}\right)$ (its existence is trivial for the identical map $w\left(z\right)\equiv z$ and for other $w\left(z\right)$ follows from Schwarz’s lemma). The relation (11) leads to an equivalent construction of the spaces $\mathbf{T}$ and $\mathrm{Fib}\left(\mathbf{T}\right)$.

In the general case the equality $w\left(z_0\right)=z_0$ must be understand in terms of the Carathéodory prime ends.

It is more convenient technically to deal with univalent functions in the complementary disk ${\mathbb{D}}^{*}$ (with a simple pole at the infinite point) and model the space $\mathbf{T}$ by the Schwarzians $S_W$ of the inverted functions $W\left(z\right)=1/w(1/z)$ for $w\in S_{\theta ,\alpha }$.

These functions form the corresponding classes ${\mathrm{\Sigma }}_{\theta ,\alpha }$ of nonvanishing univalent functions on the disk ${\mathbb{D}}^{*}$ with expansions

$$W\left(z\right)=e^{-i\theta }z+b_0+b_1{z}^{-1}+b_2{z}^{-2}+\dots ,W(1/\alpha )=1/\alpha ,$$

and $\widehat{\mathrm{\Sigma }}\left(1\right)={\bigcup }_{\theta ,\alpha }{\mathrm{\Sigma }}_{\theta ,\alpha }$.

Simple computations yield that the coefficients $a_n$ of $f\in S_{\theta ,\alpha }$ and the corresponding coefficients $b_j$ of $W\left(z\right)=1/f(1/z)\in {\mathrm{\Sigma }}_{\theta ,\alpha }$ are related by

$$b_0+e^{2i\theta }{a}_2=0,b_n+{\displaystyle \sum _{j=1}^n}{ϵ}_{n,j}{b}_{n-j}{a}_{j+1}+{ϵ}_{n+2,0}{a}_{n+2}=0,n=1,2,...,$$

where ${ϵ}_{n,j}$ are the entire powers of $e^{i\theta }$. Accordingly,

$$a_n={(-1)}^{n-1}{ϵ}_{n-1,0}{b}_0^{n-1}-{(-1)}^{n-1}(n-2){ϵ}_{1,n-3}{b}_1{b}_0^{n-3}+\mathrm{lower}\mathrm{terms}\mathrm{with}\mathrm{respect}\mathrm{to}{b}_0.$$

(12)

Note also that the Taylor coefficients ${\alpha }_n$ of Schwarzians $S_w$ are represented as polynomials of $n+2$ initial coefficients of $w\in S_{\theta ,\alpha }$ and, in view of (18), as polynomials of $n+1$ initial coefficients of the corresponding $W\in {\mathrm{\Sigma }}_{\theta ,\alpha }$.

Our next step is to lift both functionals $\tilde{J}\left(w\right)$ and $\tilde{I}\left(W\right)$ onto the Teichmüller space ${\mathbf{T}}_1$, which covers $\mathbf{T}$. Letting

$$\widehat{J}\left(\mu \right)=\tilde{J}\left(W^{\mu }\right),$$

(13)

we lift these functionals from the sets $S_{\theta ,\alpha }$ and ${\mathrm{\Sigma }}_{\theta ,\alpha }$ onto the ball $\mathrm{Belt}{\left(\mathbb{D}\right)}_1$. Then, under the indicated ${\mathbf{T}}_1$-equivalence, i.e., by the quotient map

$${\varphi }_{{\mathbf{T}}_1}:\mathrm{Belt}{\left(\mathbb{D}\right)}_1\to {\mathbf{T}}_1,\mu \to {\left[\mu \right]}_{{\mathbf{T}}_1},$$

the functional ${\tilde{J}}_n\left(W^{\mu }\right)$ is pushed down to a bounded holomorphic functional ${\mathcal{J}}_n$ on the space ${\mathbf{T}}_1$ with the same range domain.

Equivalently, one can apply the quotient map $\mathrm{Belt}{\left(\mathbb{D}\right)}_1\to \mathbf{T}$ (i.e., $\mathbf{T}$-equivalence) and compose the descended functional on $\mathbf{T}$ with the natural holomorphic map ${\iota }_1:{\mathbf{T}}_1\to \mathbf{T}$ generated by the inclusion ${\mathbb{D}}_{*}\hookrightarrow \mathbb{D}$ forgetting the puncture.

Note that since the coefficients $b_0,b_1,\dots$ of $W^{\mu }\in {\mathrm{\Sigma }}_{\theta ,\alpha }$ are uniquely determined by its Schwarzian $S_{W^{\mu }}$, the values of ${\mathcal{J}}_n$ in the points $X_1,X_2\in {\mathbf{T}}_1$ with ${\iota }_1\left(X_1\right)={\iota }_1\left(X_2\right)$ are equal.

In view of the Bers isomorphism theorem, the points of the space ${\mathbf{T}}_1$ can be regarded as the pairs $X_{F^{\mu }}=(S_{F^{\mu }},F^{\mu }\left(0\right))$, where $\mu \in \mathrm{Belt}{\left(\mathbb{D}\right)}_1$ obey ${\mathbf{T}}_1$-equivalence (hence, also $\mathbf{T}$-equivalence); hence, we get on ${\mathbf{T}}_1=\mathrm{Fib}\left(\mathbf{T}\right)$ the holomorphic functionals

$$\mathcal{J}\left(X_{F^{\mu }}\right)=\mathcal{J}(S_{F^{\mu }},t),\mathcal{I}\left(X_{F^{\mu }}\right)=\mathcal{I}(S_{F^{\mu }},t);t=F^{\mu }\left(0\right).$$

(14)

Step 2: Subharmonicity of maximal functions generated by functionals (10).

For any fixed $\theta$, the functionals (14) are defined for all $S_W\in \mathbf{T}$ and t running on some domain $D_{\theta }$ (depending on this $\theta$). This allows us to define for $t\in D_{\theta }$ the function

$${\tilde{u}}_{\theta }\left(t\right)=\underset{W^{\mu }}{\sup }\left|\mathcal{J}(S_{W^{\mu }},t)\right|,$$

(15)

taking the supremum over all $S_{W^{\mu }}\in \mathbf{T}$ admissible for any fixed given $t=W^{\mu }\left(0\right)\in D_{\theta }$ (and similarly for $\mathcal{I}\left(X_{F^{\mu }}\right)$).

We pass to the upper semicontinuous regularization of function (15).

$$u_{\theta }\left(t\right)=\underset{t^{\prime }\to t}{\mathrm{lim\; sup}}{\tilde{u}}_{\theta }\left(t^{\prime }\right).$$

In fact, we must estimate the smaller plurisubharmonic functionals arising after restriction of functionals (14) onto the complex submanifold in $\mathrm{Fib}\left(\mathbf{T}\right)$ corresponding to $G\subset \mathbf{T}$. It is filled by pairs $(S_{W^{\mu }},W^{\mu }\left(0\right))$ with $S_{W^{\mu }}\in G$ and $W^{\mu }\left(0\right)$ from some subdomain ${D^{\prime }}_{\theta }\subset D_{\theta }$. Hence, the restrictions of the moduli of both functionals (16) to this submanifold also are plurisubharmonic.

Note also that since by Lemma 2 the boundaries of domains $W^{\mu }\left({\mathbb{D}}^{*}\right)$ for all $W^{\mu }\left(z\right)\in {\mathrm{\Sigma }}^0$ are located in the disk ${\displaystyle \left\{\right|W-b_0|\le |a_{2,\theta }^0|\le 2\}}$ and both functionals $J,I$ are rotationally invariant, the constructed domain ${D^{\prime }}_{\theta }$ s a disk ${\mathbb{D}}_{\alpha }$ of some radius

$$\alpha \le 2|a_{2^0}|=2\underset{\theta }{\max }\left|a_{2,\theta }^0\right|$$

(of course, all quantities in the last relation also depend on p and q, which are fixed). Now, the crucial step in the proof is to establish that the function (15) inherits the subharmonicity of $\mathcal{J}$. This is given by the following lemma, which is a consequence of the corresponding lemma in [5] related to functionals on the whole space $\mathbf{T}$.

Lemma 5.

The function $u_{\theta }\left(t\right)$ is subharmonic in the indicated disk ${\mathbb{D}}_{2|a_{2^0}|}$.

The proof of the indicated lemma in [5] is very complicated. In view of its importance, we outline here the main arguments applied in [5], omitting the details.

First, we apply a weak approximation of the underlying space $\mathbf{T}$ (and simultaneously of the space ${\mathbf{T}}_1$) by finite dimensional Teichmüller spaces of the punctured spheres in the topology of locally uniform convergence on $\mathbb{C}$. Take the set of points

$$E=\{e^{\pi si/2^n},s=0,1,\dots ,2^{n+1}-1;n=1,2,\dots \},$$

which is dense on ${\mathbb{S}}^1$, and consider the punctured spheres

$$X_m=\widehat{\mathbb{C}}\setminus \{e^{\pi si/2^n},s=0,1,\dots ,2^{n+1}-1\},m=2^{n+1},$$

and their universal holomorphic covering maps $g_m:\mathbb{D}\to X_m$ normalized by $g_m\left(0\right)=0,g_m^{\prime }\left(0\right)>0$.

The radial slits from the infinite point to all the points $e^{\pi si/2^n}$ form a canonical dissection $L_m$ of $X_m$ and define the simply connected surface $X_m^{\prime }=X_m\setminus L_m$. Any covering map $g_m$ determines a Fuchsian group ${\mathrm{\Gamma }}_m$ of covering transformations acting discontinuosly in both disks $\mathbb{D}$ and ${\mathbb{D}}^{*}$. The (open) fundamental polygon $P_m$ of ${\mathrm{\Gamma }}_m$ in $\mathbb{D}$ corresponding to the dissection $L_m$ is a regular circular $2^{n+1}$-gon centered at the origin, which can be chosen to have a vertex at 1. These polygons entirely increase and exhaust the disk $\mathbb{D}$, and the restriction of $g_m$ to $P_m$ is univalent.

Similarly, we take in the complementary disk ${\mathbb{D}}^{*}$ the mirror polygons $P_m^{*}$ and the covering maps $g_m^{*}\left(z\right)=1/\overline{g_m(1/\overline{z})}$, which define the mirror surfaces $X_m^{*}$.

Now, we approximate the maps $F^{\mu }\in {\mathrm{\Sigma }}_{\theta }\left(1\right)$ by homeomorphisms $F^{{\mu }_m}$ having in $\mathbb{D}=\left\{\right|z|<1\}$ the Beltrami coefficients

$${\mu }_m={\left[g_m\right]}_{*}\mu :=(\mu \circ g_m)\overline{g_m^{\prime }}/g_m^{\prime },n=1,2,\dots .$$

Each $F^{{\mu }_m}$ is again k-quasiconformal and compatible with the group ${\mathrm{\Gamma }}_m$. As $m\to \infty$, the coefficients ${\mu }_m$ are convergent to $\mu$ almost everywhere on $\mathbb{C}$; thus, the maps $F^{{\mu }_m}$ are convergent to $F^{\mu }$ uniformly in the spherical metric on $\widehat{\mathbb{C}}$. Note also that ${\mu }_m$ depend holomorphically on $\mu$ as elements of $L_{\infty }$ and on $t=F^{\mu }\left(0\right)$.

This construction and the features of extremal quasiconformal maps allows one to first define the corresponding subharmonic functions for the Schwarzians $S_{W^{g_m}}$ in $(m-3)$-dimensional subdomain of $\mathbf{T}$, and then investigate the limit function (15) arising as $m\to \infty$.

To establish that the domain of function (15) is the maximal disk ${\mathbb{D}}_{2|a_{2^0}|}$ (in [5] this is the disk ${\mathbb{D}}_4$), we apply the following symmetrization. Take the increasing unions of the quotient spaces

$${\mathcal{T}}_s={\displaystyle \bigcup _{j=1}^s}{\widehat{\mathrm{\Sigma }}}_{{\theta }_j}^0/\sim ={\displaystyle \bigcup _{j=1}^s}\{(S_{W_{{\theta }_j}},W_{\theta }^{\mu }\left(0\right))\}\simeq {\mathbf{T}}_1\cup \dots \cup {\mathbf{T}}_1,$$

(16)

where ${\theta }_j$ run over a dense subset $\Theta \subset [-\pi ,\pi ]$, the equivalence relation ∼ means ${\mathbf{T}}_1$-equivalence on a dense subset ${\widehat{\mathrm{\Sigma }}}^0\left(1\right)$ in the union $\widehat{\mathrm{\Sigma }}\left(1\right)$ formed by univalent functions $W_{{\theta }_j}\left(z\right)=e^{-i{\theta }_j}z+b_0+b_1{z}^{-2}+\dots$ on ${\mathbb{D}}^{*}$ with quasiconformal extension to $\widehat{\mathbb{C}}$ satisfying $W_{{\theta }_j}\left(1\right)=1$, and

$${\mathbf{W}}_{\theta }^{\mu }\left(0\right):=(W_{{\theta }_1}^{{\mu }_1}\left(0\right),\dots ,W_{{\theta }_s}^{{\mu }_s}\left(0\right)).$$

The Beltrami coefficients ${\mu }_j\in \mathrm{Belt}{\left(\mathbb{D}\right)}_1$ are chosen here independently. The corresponding collection

$$\beta =({\beta }_1,\dots ,{\beta }_s)$$

of the Bers isomorphisms

$${\beta }_j:\{(S_{W_{{\theta }_j}},W_{{\theta }_j}^{{\mu }_j}\left(0\right))\}\to \mathrm{Fib}\left(\mathbf{T}\right)$$

determines a holomorphic surjection of the space ${\mathcal{T}}_s$ onto $\mathrm{Fib}\left(\mathbf{T}\right)$. The function (15) is determined by

$$u\left(t\right)=\underset{\Theta }{\sup }{u}_{{\theta }_s}\left(t\right),$$

where $u_{{\theta }_s}$ is obtained by maximization of type (15) over ${\mathcal{T}}_s$. This completes the proof of Lemma 4 for f running over the whole space $\mathbf{T}$.

To finish the proof of Lemma 4 for our case, observe that since, by assumption, the given functions f from the space ${\mathbf{B}}_{p,q}$ form a submanifold in the space $\mathbf{T}$, the restrictions of the original functional $\left|\mathcal{J}\right(S_{W^{\mu }},t\left)\right|$ to this manifold and to corresponding values of $t=W^{\mu }\left(0\right)$ also are plurisubharmonic, which implies similar to above the subharmonic maximal function

$$u_{\theta }\left(t\right)=\underset{W^{\mu }}{\sup }\left|\mathcal{J}(S_{W^{\mu }},t)\right|$$

on the corresponding disk ${\mathbb{D}}_{2|a_{2^0}|}$.

Step 3: Connection between the extremals $w_0$ and $f_0$.

The next crucial point in the proof of Theorem 1 is to establish that the function $w_0$ with the maximal radius $2|a_2^0|$ is naturally connected with the extremal function $f_0$ maximizing the coefficient $|c_1|$, namely, $f_0=S_{w_0}$ up to a constant factor.

We need to prove that

$$S_{w_0}^{\prime }\left(0\right)=c_1^0\ne 0.$$

(17)

This yields that the zero set of the functional $J_1\left(f\right)=c_1$ on $\widehat{S}\left(1\right)$ is separated from the set of rotations (17) of the function $w_0$, and therefore the corresponding function (14), constructed by the maximization of functional $|J_1\left(f\right)|=|c_1|$, is logarithmically subharmonic on the disk ${\mathbb{D}}_{2|a_2^0|}$ (and attaining its maximum on its boundary).

There are a possible two cases: either $f_0\left(z\right)$ is a linear function $a+bz$ or its series expansion contains more powers. In the first case, the inequality (17) holds trivially.

So, consider the second case, for which one can apply Lemma 3 (with $n=1$).

Fix $M<\infty$ and let $B_M$ be the connected subset of $L_{2m,\delta }$ containing the functions $f\left(z\right)$ with ${\parallel f\parallel }_{\infty }\le M$.

For each M we have the subharmonic function

$$u_M\left(t\right)=\underset{B_M}{\sup }\left\{\right|\mathcal{J}(S_{W^{\mu }},t)|.$$

on the corresponding disk ${\mathbb{D}}_{2|a_{2,M}^0|}\subset {\mathbb{D}}_{2|a_2^0|}$. These functions $u_M\left(t\right)$ are monotone increasing with M, i.e., for $M^{\prime }>M$, $u_{M^{\prime }}\left(t\right)\ge u_M\left(t\right)$, and similarly for the corresponding radii $2|a_{2,M}^0|$.

The corresponding collection of $f_M=S_{w_{0,M}}$ with maximal $|c_{1,M}|$ on $B_M$ is weakly compact in $L_{2m,\delta }$ (in the weak topology on $L_{2m,\delta }$ defined by linear functionals) as well as in the topology of locally uniform convergence of continuous functions on the unit disk. Hence, as $M\to \infty$, the maximal values $c_{1,M}^0$ and $a_{2,M}^0$ on $B_M$ approach the corresponding maximal values $|c_1^0|$ and $|a_2^0|$ of the coefficients $c_1$ and $a_2$ on $L_{2m,\delta }$. The limit subharmonic function $u\left(t\right)={\lim }_{M\to \infty }{u}_M\left(t\right)$ must coincide with (15).

Now, fix a sufficiently small $\epsilon =\epsilon \left(M\right)>0$ and choose for the extremal on $B_M$ function

$$f_{0,M}\left(z\right)=S_{w_{0,M}}\left(z\right)=c_M^0+c_1^{0}z+\dots$$

a large $M>\parallel f_{0,M}{\parallel }_{2m}$, so that

$$|c_{1,M}^0|\ge |c_1^0|-\epsilon ,|a_{2,M}^0|\ge |a_2^0|-\epsilon .$$

Since the image $f_{0,M}\left(\mathbb{D}\right)$ is a proper subset of the disk ${\mathbb{D}}_M=\left\{\right|w|<M\}$, one can vary arbitrarily the extremal value $|c_{1,M}^0|$ using Lemma 3, which provides $O\left(\epsilon \right)$-quasiconformal automorphisms $h_{\epsilon }$ of $\widehat{\mathbb{C}}$ conformal in the domain $f_{0,M}\left(\mathbb{D}\right)$ and such that $\parallel S_{h_{\epsilon }\circ f_{0,M}}{\parallel }_{2m}={\parallel S_{f_{0,M}}\parallel }_{2m}+O\left(\epsilon \right)$, and

$$|a_2(h_{\epsilon }\circ w_{0,M})|=|a_2\left(w_{0,M}\right)|,|c_1(h_{\epsilon }\circ f_{0,M})|=|c_{1,M}^0|+O\left(\epsilon \right)>\left|c_{1,M}^0\right|.$$

Letting $M\to \infty$, one obtains that the circles $\left\{\right|t|=2|a_{2,M}^0|+O\left(\epsilon \right)\}$ approach from inside the circle $\left\{\right|t|=2|a_2^0\left|\right\}$, and the limit maximal function (14) for functional $I\left(f\right)|$ is subharmonic on the disk ${\mathbb{D}}_{2|a_2^0|}$ determined by $w_0\left(z\right)$. This provides the estimate (5).

Step 4: Proof of estimate (6).

To obtain the estimate (6), one must consider instead of (15) the functional

$$I_n\left(f\right)=\max \left(\right|J\left(f\right)|,|J\left(f_n\right)\left|\right)$$

with $f_n\left(z\right)=f\left(z^n\right)$. Similar to the previous case of $J_1\left(f\right)$, one can lift both components of $I_n\left(f\right)$ onto the Teichmüller spaces $\mathbf{T}$ and ${\mathbf{T}}_1$ and, applying the above lemmas, construct for $I_n$ the corresponding dominant circularly symmetric subharmonic function (15) defined on the same disk ${\mathbb{D}}_{2|a_2^0|}$. This implies the bound (6).

Step 5: Proof of part $\left(c\right)$.

As was mentioned above, in the case of nonvanishing functions $f\in L_{q,\delta }$, Lemma 3 is valid for all $q\ge 2$. Another crucial point is that the need for holomorphy and subharmonicity openness of the domain of function is ensured by Lemma 4. It allows one to apply the same arguments as in the previous steps, which provides the assertion $\left(c\right)$. This completes the proof of Theorem 1. □

7. Example

Consider the balls $B_r={\{\parallel f\parallel }_{2,\sigma }<r\}$ in the Hilbert space $A_{2,\sigma }\left(\mathbb{D}\right)$ of holomorphic functions

$$f\left(z\right)=c_0+c_{1}z+c_2{z}^2+\dots ,\left|z\right|<1,$$

with norm

$${\parallel f\parallel }_{2,\sigma }={\left({\displaystyle \frac{1}{2\pi }}\underset{\mathbb{D}}{\int \int }{\sigma \left(\right|z\left|\right)\left|f\left(z\right)\right|}^{2}dxdy\right)}^{1/2},$$

where the weight $\sigma$ is a radial continuous function, and use its orthonormal basis

$$f_n\left(z\right)={\lambda }_n{z}^n,n=0,1,\dots$$

The Taylor coefficients $c_n$ of $f\in A_{2,\sigma }$ are the products $c_n={\lambda }_n{g}_n$ of their Fourier coefficients $g_n$ with some factors ${\lambda }_n^{\prime }$ depending on $\sigma$, and by Parseval’s equality

$${\parallel f\parallel }_{2,\sigma }={\displaystyle \sum _0^{\infty }}{\lambda }_n^2{g}_n^2.$$

Take the ball $B_{r_0}$ of a radius $r_0$ in $A_{2,\sigma }$, so that the $\mathbf{B}$-norm of any its point is less than 2 (hens, any of its point is the Schwarzian of a univalent $w_0$ in $\mathbb{D}$) and choose $\sigma \left(r\right)$ so that

$$g_1={\displaystyle \frac{1}{2\pi }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}}{r}^2\sigma \left(r\right)dr>{\displaystyle \frac{r_0}{{\lambda }_1}}.$$

Then we have in (6) the indicated case $|c_n^0|\le |c_1^0|$ for all $n\ge 2$, and the maximal value of each $|c_n|$ equals $c_1$ and is attained on $B_{r_0}$ on the function $f_0\left(z\right)=c_1{z}^n$ (up to a factor).

Theorem 1 yields that the function $w_0\left(z\right)$ with $S_{w_0}=c_1{z}^n{\lambda }_n$ maximizes simultaneously both coefficients $a_2$ and $c_1$ on this class, and this function also is extremal for all $a_n,n\ge 3$.

To determine explicitly the Taylor coefficients $a_n$ of univalent $w\left(z\right)$, whose Schwarzians run over the closed ball $\overline{B_{r_0}}$, one has to find the ratios $w={\eta }_1/{\eta }_2$ of two independent solutions of the linear differential Equation (3) with a given holomorphic f. In our case, this equation assumes the form

$${\eta }^{″}+c{z}^p\eta =0.$$

(18)

Such equations closely relate to special functions (see, e.g., [31]). In particular, in this case the extremal univalent function $w_0$ maximizing $a_2$ is determined from the equation of type (15) (for $p=1$) as a linear combination of cylindrical functions (this equation corresponds to the first case in Step 3).

8. A Generalization of Theorem 1 and Open Problems

8.1. Generalization

It is important for applications to find more general spaces admitting the assertions similar to Theorem 1. A simple general theorem of such type is the following one.

Let X be a Banach space embedded in ${\mathbf{B}}_2$, i.e., such that

$${\parallel f\parallel }_X\le \mathrm{const}{\parallel f\parallel }_{{\mathbf{B}}_2}.$$

Pick the subdomains $G\subset X$ whose boundaries have the common points with $\partial \mathbf{T}$.

Theorem 3.

Consider for a subdomain $G\subset X$ the univalent solutions $w\left(z\right)$ generated by $f\in G$ via Equations (1) or (3), and let the Schwarzian $S_{w_0}$ of function $w_0$ with maximal $|a_2|$ on $\overline{G}$ has nonzero first coefficient $c_1^0$. Then,

$$|c_1^0|=\underset{f\in G}{\sup }\left|c_1\left(f\right)\right|,$$

and all $f\in \overline{G}$ obey the estiamte (6).

The proof of this theorem follows the above lines. One problem is to find such spaces X explicitly.

8.2. Problems

An intriguing unsolved question connected with the above theorems is how to extend the variational Lemma 3 to holomorphic functions which belong to collections in more general Banach (in particular, Hilbert) spaces.

This would open a way to solving the Hummel–Scheinberg–Zalcman problem on estimating the coefficients of nonvanishing holomorphic functions from arbitrary Banach spaces stated in [2], as well as also having other interesting applications.

Another interesting problem is to find the Banach submanifolds in $\mathbf{T}$ obeying, similar to spaces ${\mathbf{B}}_{p,q}$, the property $c_1\left(S_{w_0}\right)\ne 0$.