Extremality of Koebe’s Function

Abstract

The remarkable Koebe function is the (unique) extremal of many important distortion functionals in geometric function theory. This paper provides a complete characterization of such functionals.

1. Introductory Remarks: Main Theorem

1.1. Preamble

It is well known that many holomorphic functionals $J\left(f\right)$ on the canonical class S of univalent functions

$$f\left(z\right)=z+a_2{z}^2+\cdots +a_n{z}^n+\dots$$

(i.e., with $f\left(0\right)=0,f^{\prime }\left(1\right)=1$) in the unit disk $\mathbb{D}=\left\{\right|z|<1\}$ are maximized by the Koebe function

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

(1)

which maps the unit disk onto the complement of the ray

$$w=-t{e}^{-i\theta },{\displaystyle \frac{1}{4}}\le t\le \infty$$

(see, e.g., [1,2,3,4,5,6,7,8,9,10,11] and the references cited there). Especially, this holds for many important coefficient functionals

$$J\left(f\right)=J(a_{m_1},\dots ,a_{m_s})$$

(2)

depending on the distinguished finitely Taylor coefficients $a_{m_j}$. Such functionals arise and play a crucial role in many mathematical and physical applications of geometric complex analysis. A very intriguing and important old problem is to characterize the functionals possessing the indicated extremality intrinsically.

We shall assume that

$$2<a_{m_1}<\cdots <a_{m_s}<\infty .$$

The left-hand inequality is caused by the fact that the coefficient $a_2$ can be included in the Cauchy initial conditions

$$w\left(0\right)=0,w^{\prime }\left(0\right)=1,w^{″}\left(0\right)=a_2,$$

uniquely determining a univalent function $w\left(z\right)$ as the solution of the nonlinear Schwarz differential equation

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

with an appropriate holomorphic $\phi$ on $\mathbb{D}$; this distinguishes a subclass of S.

Our aim is to describe which coefficient functionals (2) are maximized by function (1), giving a solution to the indicated problem.

It is natural to assume that the functionals considered here are rotationally homogeneous, which means that for any $f\left(z\right)\in S$ and its rotations

$$f_{\alpha }\left(z\right)=e^{-i\alpha }f\left(e^{i\alpha }z\right),\alpha \in [-\pi ,\pi ],$$

we have $J\left(f_{\alpha }\right)=J\left(f\right)e^{ip\alpha }$ with some $p\in \mathbb{N}$.

1.2. The Associated Functional

For any $f\in S$, its inverted function

$$F_f\left(z\right)=1/f(1/z)=z+b_0+b_1{z}^{-1}+\dots$$

(3)

is univalent and zero free on the complementary disk ${\mathbb{D}}^{*}=\{z\in \widehat{\mathbb{C}}=\mathbb{C}\cup \{\infty \}:\left|z\right|>1\}$, with a simple pole at $z=\infty$ (so $\widehat{\mathbb{C}}$-holomorphic on ${\mathbb{D}}^{*}$). The class of all univalent functions on ${\mathbb{D}}^{*}$ with expansions (3) is denoted by $\Sigma$.

Let $S_Q$ and ${\Sigma }_Q$ denote the subclasses of S and $\Sigma$ formed by functions with quasiconformal extensions (onto ${\mathbb{D}}^{*}$ and $\mathbb{D}$, respectively). These subclasses are dense in the weak topology generated by convergence in the spherical metric on $\widehat{\mathbb{C}}$.

The coefficient $a_n$ of f and the corresponding coefficient $b_j$ of $F_f$ are related via

$$b_0+a_2=0,b_n+\sum _{j=1}^n{b}_{n-j}{a}_{j+1}+a_{n+2}=0,n=1,2,\dots ,$$

which successively provides the representations of $a_n$ by $b_j$:

$$a_n={(-1)}^{n-1}{b}_0^{n-1}-{(-1)}^{n-1}(n-2)b_1{b}_0^{n-3}+\mathrm{lower}\mathrm{terms}\mathrm{with}\mathrm{respect}\mathrm{to}{b}_0.$$

(4)

These relations transform the initial functional $J\left(f\right)$ into a coefficient functional $\tilde{J}\left(F\right)$ on $\Sigma$, depending on the corresponding coefficients $b_j$, which will be regarded as associated with J.

1.3. Remarks on the Beltrami and Schwarz Equations

Any $F\in {\Sigma }_Q$ is the restriction to ${\mathbb{D}}^{*}$ of a generalized homeomorphic solution of the Beltrami equation ${\partial }_{\overline{z}}w=\mu {\partial }_{z}w$ on the complex plane $\mathbb{C}$, where the partial derivatives are distributional and the Beltrami coefficient $\mu$ (or complex dilatation of the map w) belongs to the unit ball

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

To have uniqueness of solution, one must add the third normalization condition, for example, $w(\infty )=\infty$ or $w\left(1\right)=1$. This also yields the compactness of maps with ${\parallel \mu \parallel }_{\infty }\le k<1$, holomorphic dependence with $\mu (·;\zeta )$ from a complex parameter $\zeta$ running over some Banach domain, etc.

We shall denote the solutions with a complete normalization by $w^{\mu }\left(z\right)$.

Another important Möbius invariant of a function $F\in \Sigma$ is its Schwarzian derivative $S_F$ defined by

$$S_F\left(z\right)={\left({\displaystyle \frac{F^{″}\left(z\right)}{F^{\prime }\left(z\right)}}\right)}^{\prime }-{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{F^{″}\left(z\right)}{F^{\prime }\left(z\right)}}\right)}^2(z\in {\mathbb{D}}^{*}).$$

We mention the chain rule

$$S_{F_1\circ F}\left(z\right)=(S_{F_1}\circ F)F^{\prime }{\left(z\right)}^2+S_F\left(z\right)$$

giving, for the Möbius maps $w=\gamma \left(z\right)$, the equality

$$S_{F_1\circ \gamma }\left(z\right)=(S_{F_1}\circ \gamma ){\gamma }^{\prime }{\left(z\right)}^2,S_{\gamma \circ F}\left(z\right)=S_F\left(z\right).$$

Hence, each $S_F\left(z\right)$ can be regarded as a quadratic differential $\phi =S_F\left(z\right)d{z}^2$ on ${\mathbb{D}}^{*}$. Either quantity: the Beltrami coefficient ${\mu }_F={\partial }_{\overline{z}}F/{\partial }_z$ and the Schwarzian $S_F$ defined the map F up to a Möbius transformation of $\widehat{\mathbb{C}}$.

For every locally univalent function $w\left(z\right)$ on a simply connected hyperbolic domain $D\subset \widehat{\mathbb{C}}$, its Schwarzian derivative belongs to the complex Banach space $\mathbf{B}\left(D\right)$ of hyperbolically bound holomorphic functions on D with the norm

$${\parallel \phi \parallel }_{\mathbf{B}\left(D\right)}=\underset{D}{sup}{\lambda }_D^{-2}\left(z\right)\left|\phi \left(z\right)\right|,$$

where ${\lambda }_D\left(z\right)\left|dz\right|$ is the hyperbolic metric on D of Gaussian curvature $-4$; hence, $\phi \left(z\right)=O\left(z^{-4}\right)$ as $z\to \infty$ if $\infty \in D$. In particular,

$${\lambda }_{\mathbb{D}}\left(z\right)={1/(1-|z|}^2),{\lambda }_{{\mathbb{D}}^{*}}\left(z\right)=1/{\left(\right|z|}^2-1).$$

1.4. A Distinguished Subclass of $\Sigma$

For each $F\left(z\right)=z+b_0+b_1{z}^{-1}+\cdots \in \Sigma$, we define a complex homotopy

$$F_t\left(z\right)=tF\left({\displaystyle \frac{z}{t}}\right)=z+b_{0}t+b_1{t}^2{z}^{-1}+b_2{t}^3{z}^{-2}+\dots :{\mathbb{D}}^{*}\times \mathbb{D}\to \widehat{\mathbb{C}}$$

of this function to the identity map. Then

$$S_{F_t}\left(z\right)=t^{-2}{S}_F\left(t^{-1}z\right)$$

and, moreover, the map $h_F:t\mapsto S_{F_t}$ is holomorphic as a function $\mathbb{D}\to \mathbf{B}$. It determines the homotopy disk $\mathbb{D}\left(F\right)=\left\{F_t\right\}$, which is holomorphic at the noncritical points of $h_F$. These disks foliate the set ${\Sigma }_Q$.

The corresponding homotopy of functions from S is given by $f(z,t)=t^{-1}f\left(tz\right)=z+a_{2}t+\dots$; so $J\left(f(·,t)\right)=t^{2n-2}J\left(f\right)$.

Each homotopy map $F_t$ admits k-quasiconformal extension to the whole sphere $\widehat{\mathbb{C}}=\mathbb{C}\cup \{\infty \}$ with $k\le {\left|t\right|}^2$. The bound $k\left(F_t\right)\le {\left|t\right|}^2$ is sharp and occurs only for the maps

$$F_{b_0,b_1;1}\left(z\right)=z+b_0+b_1{z}^{-1},\left|b_1\right|=1,$$

whose homotopy maps

$$F_{b_0,b_1;t}\left(z\right)=z+b_{0}t+b_1{t}^2{z}^{-1}$$

(5)

have the affine extensions ${\widehat{F}}_{b_0,b_1;t}\left(z\right)=z+b_{0}t+b_1{t}^2\overline{z}$ onto $\mathbb{D}$.

Due to Strebel’s frame mapping condition [12], the extremal extensions ${\widehat{F}}_t$ of any homotopy functions $F_t$ with $\left|t\right|<1$ is of Teichmüller type, i.e., with the Beltrami coefficient of the form

$${\mu }_{{\widehat{F}}_t}\left(z\right)=\tau \left(t\right)\left|\psi \left(z\right)\right|/\psi \left(z\right),$$

where $\psi$ is a holomorphic function from $L_1\left(\mathbb{D}\right)$ (and unique).

We divide every homotopy function $F_t$ of $F=F_f$ into two parts

$$F_t\left(z\right)=z+b_{0}t+b_1{t}^2{z}^{-1}+b_2{t}^3{z}^{-2}+\cdots =F_{b_0,b_1;t}\left(z\right)+h(z,t),$$

where $F_{b_0,b_1;t}$ is the map (5) with $b_0,b_1$ coming from F. For a sufficiently small $\left|t\right|$, the remainder h is estimated by $h(z,t)=O\left(t^3\right)$ uniformly in z for all $\left|z\right|\ge 1$.

Then the Schwarzian derivatives of $F_t$ and $F_{b_0,b_1;t}$ are related by

$$S_{F_t}\left(z\right)=S_{F_{b_0,b_1;t}}\left(z\right)+\omega (z,t),$$

where the remainder $\omega$ is uniquely determined by the chain rule

$$S_{w_1\circ w}\left(z\right)=(S_{w_1}\circ w){\left(w^{\prime }\right)}^2\left(z\right)+S_w\left(z\right),$$

and is estimated in the norm of $\mathbf{B}$ by ${\parallel \omega (·,t)\parallel }_{\mathbf{B}}=O\left(t^3\right),t\to 0$; this estimate is uniform for $\left|t\right|<t_0$ (cf., e.g., [3,13]).

All functions $F_{b_0,b_1;t}$ with

$$|b_0|\le 2,|b_1|\le 1,|t|\le 1.$$

(6)

are univalent on the disk ${\mathbb{D}}^{*}$ (but can vanish there) and, if $\left|t\right|<1$, have the affine extensions onto $\mathbb{D}$. For such functions, their homotopy disk $\mathbb{D}\left(F\right)=\left\{F_t\right\}$ coincides with the extremal disk $\mathbb{D}\left(\psi \right)=\{t{\mu }_0:t\in \mathbb{D}\}\subset Belt{\left(\mathbb{D}\right)}_1$; hence, the action of the functional $\tilde{J}$ on extremal disks of functions $F_{b_0,b_1;t}$ is rotationally symmetric with respect to $t\in \mathbb{D}$.

We call the values $b_0$ and $b_1$ admissible if they are the initial coefficients of some function from ${\Sigma }_Q$ (these values satisfy (6)). The collection of all such $F_t$ with $\left|t\right|<1$ will be denoted by ${\Sigma }_{af}$. To have compactness, we shall also use the closure of this set with respect to locally uniform convergence on ${\mathbb{D}}^{*}$.

1.5. Main Results

It is convenient to present our main results in terms of the associated functional.

Theorem 1.

The Koebe function ${\kappa }_{\theta }\left(z\right)$ is (a unique) extremal of a rotationally homogeneous coefficient functional $J\left(f\right)$ if and only if the associated functional $\tilde{J}\left(F\right)$ satisfies

$$\underset{\Sigma }{max}\tilde{J}\left(F\right)=\underset{f\in S}{max}|\tilde{J}\left(F_f\right)|=\underset{{\Sigma }_{af}}{sup}\left|\tilde{J}\left(F_{b_0,b_1;t}\right)\right|;$$

(7)

In other words, the maximal value of the associate to J functional $\tilde{J}$ on $\Sigma$ must be attained on the distinguished subset ${\Sigma }_{af}$ of functions $F\in \Sigma$, admitting affine extensions to $\mathbb{D}$.

As a simple consequence, the following is useful:

Corollary 1.

The Koebe function cannot be an extremal of any coefficient functional $J\left(f\right)$ whose associated functional $\tilde{J}\left(f\right)$ satisfies

$$\underset{\Sigma }{max}|\tilde{J}\left(F\right)|>\underset{{\Sigma }_{af}}{sup}\left|\tilde{J}\left(F\right)\right|.$$

Theorem 1 is extended straightforwardly to the general holomorphic functionals (2) on S with a holomorphic function J on an appropriate bounded domain $G\subset {\mathbb{C}}^s$ containing the distinguished coefficients $a_{m_1},\dots ,a_{m_s}$. It is clear that the inequality indicated in Corollary 1 holds for most of the holomorphic functionals on S, because the affine maps form a very sparse set among arbitrary quasiconformal expansions. So, most of the extremal functions are different from the Koebe function.

Simple explicit examples of such functionals are generated by polynomials

$$\tilde{J}\left(F\right)=\sum _3^p{C}_j{b}_j^N(p>3,N>3)$$

(8)

on $\Sigma$, applying relation (4). For any such $\tilde{J}$, the function ${\kappa }_{\theta }$ lies in the zero set of the corresponding functional J on S. More generally, one can add to polynomial (8) the sums $C_0{b}_0^{q_0}+C_1{b}_1^{q_2}$ with sufficiently small $C_0,C_1$.

Theorem 1 implicitly embraces many classical distortion results of geometric function theory, estimating the coefficients. The method developed in [4,5,6] implies that in these problems the extremal functions can be obtained by maximization of the given functionals along the set ${\Sigma }_{af}$, and these functions obey (7).

2. Digression to Teichmüller Spaces

First, we briefly recall the underlying results from Teichmüller space theory, which play a crucial role in the proof of Theorem 1; for details see, e.g., [14,15,16]. This theory is intrinsically connected with univalent functions having quasiconformal extensions onto $\widehat{\mathbb{C}}$.

Quasiconformality requires three normalization conditions to have uniqueness, compactness, holomorphic dependence on parameters, etc. It is technically more convenient to deal with functions from ${\Sigma }_Q$.

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

The canonical complex Banach structure on $\mathbf{T}$ is defined by factorization of the ball $Belt{\left(\mathbb{D}\right)}_1$ letting ${\mu }_1,{\mu }_2\in Belt{\left(\mathbb{D}\right)}_1$ be equivalent if the corresponding quasiconformal maps $w^{{\mu }_1},w^{{\mu }_2}$ 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 equivalence classes ${\left[w^{\mu }\right]}_{\mathbf{T}}$ are in one-to-one correspondence with the Schwarzians $S_{w^{\mu }}$ of restrictions $w^{\mu }$ to ${\mathbb{D}}^{*}$.

These Schwarzians range over a bounded domain in the space $\mathbf{B}=\mathbf{B}\left(\mathbb{D}\right)$, which models the space $\mathbf{T}$. It is located in the ball $\{\parallel \phi {\parallel }_{\mathbf{B}}<6\}$ and contains the ball $\{\parallel \phi {\parallel }_{\mathbf{B}}<2\}$. In this model, the Teichmüller spaces of all hyperbolic Riemann surfaces are contained in $\mathbf{T}$ as its complex submanifolds.

The factorizing projection ${\varphi }_{\mathbf{T}}\left(\mu \right)=S_{w^{\mu }}:Belt{\left(\mathbb{D}\right)}_1\to \mathbf{T}$ is a holomorphic map from $L_{\infty }\left(\mathbb{D}\right)$ to $\mathbf{B}$. This map is a split submersion, which means that ${\varphi }_{\mathbf{T}}$ has local holomorphic sections (see, e.g., [13,15,17,18]).

Both equations $S_w=\phi$ and ${\partial }_{\overline{z}}w=\mu {\partial }_{z}w$ (on ${\mathbb{D}}^{*}$ and $\mathbb{D}$, respectively) determine their solutions up to a Möbius transformation of $\widehat{\mathbb{C}}$.

The following lemma provides a somewhat different normalization of quasiconformally extendable functions, which also ensures (as was mentioned above) the needed uniqueness of solutions, their holomorphic dependence on complex parameters, etc.

Lemma 1

([5]). For any Beltrami coefficient $\mu \in Belt{\left({\mathbb{D}}^{*}\right)}_1$ 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

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

(9)

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 $\mathbb{D}$, normalized by $w\left(0\right)=0,w^{\prime }\left(0\right)=1,w\left(1\right)=1$ and with more general normalization

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

Note that for $\mu \left(z\right)=0$ (almost everywhere on ${\mathbb{D}}^{*}$) the corresponding solution $w^{\mu }\left(z\right)$ with $w\left(0\right)=0,w^{\prime }\left(0\right)=e^{i\theta },w\left(1\right)=1$ is the elliptic Móbius map

$$w=e^{-i\theta }z/((e^{-i\theta }-1)z+1)$$

with the fixed points 0 and 1, which equals the identity map when $\theta =0$.

The points of Teichmüller space ${\mathbf{T}}_1=Teich\left({\mathbb{D}}_{*}\right)$ of the punctured disk ${\mathbb{D}}_{*}=\mathbb{D}\setminus \left\{0\right\}$ are the classes ${\left[\mu \right]}_{{\mathbf{T}}_1}$ of ${\mathbf{T}}_1$-equivalent Beltrami coefficients $\mu \in Belt{\left(\mathbb{D}\right)}_1$, which means that the corresponding quasiconformal automorphisms $w^{\mu }$ of the unit disk coincide on both boundary components of ${\mathbb{D}}_{*}$ (the unit circle ${\mathbb{S}}^1=\left\{\right|z|=1\}$ 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.

Namely, the disk ${\mathbb{D}}_{*}$ is conformally equivalent to the factor $\mathbb{D}/\Gamma$, where $\Gamma$ is 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 $(-1,1)$-measurable forms $\tilde{\mu }d\overline{z}/dz$ in $\mathbb{D}$ with respect to $\Gamma$, i.e., via $(\tilde{\mu }\circ \gamma )\overline{{\gamma }^{\prime }}/{\gamma }^{\prime }=\tilde{\mu },\gamma \in \Gamma$, forming the Banach space $L_{\infty }(\mathbb{D},\Gamma )$.

We extend these $\tilde{\mu }$ by zero to ${\mathbb{D}}^{*}$ and consider the unit ball $Belt{(\mathbb{D},\Gamma )}_1$ of $L_{\infty }(\mathbb{D},\Gamma )$. Then the corresponding Schwarzians $S_{w^{\tilde{\mu }}|{\mathbb{D}}^{*}}$ belong to $\mathbf{T}$. Moreover, ${\mathbf{T}}_1$ is canonically isomorphic to the subspace $\mathbf{T}(\Gamma )=\mathbf{T}\cap \mathbf{B}(\Gamma )$, where $\mathbf{B}(\Gamma )$ consists of elements $\phi \in \mathbf{B}$ satisfying $(\phi \circ \gamma ){\left({\gamma }^{\prime }\right)}^2=\phi$ in ${\mathbb{D}}^{*}$ for all $\gamma \in \Gamma$.

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

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

over the universal space $\mathbf{T}$ with holomorphic projection $\pi (\psi ,z)=\psi$ (see [14]).

This fiber space is a bounded hyperbolic domain in $\mathbf{B}\times \mathbb{C}$ 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 $\mathcal{F}\left(\mathbf{T}\right)$ is induced by the inclusion map $j:{\mathbb{D}}_{*}\hookrightarrow \mathbb{D}$, forgetting the puncture at the origin via

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

(10)

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 $X_0\setminus \left\{x_0\right\}$ and implies that $\mathbf{T}(X_0\setminus \left\{x_0\right\})$ is biholomorphically isomorphic to the Bers fiber space $Fib\left(\mathbf{T}\left(X_0\right)\right)$ over $\mathbf{T}\left(X_0\right)$.

The spaces $\mathbf{T}$ and ${\mathbf{T}}_1$ can be weakly (in the topology generated by the spherical metric on $\widehat{\mathbb{C}}$) approximated 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 $z_j\in \mathbb{C}\setminus \{0,1\}$ (see, e.g., [4]).

Fix a collection ${\mathbf{z}}^0=(0,1,z_1^0,\dots ,z_{n-3}^0,\infty )$ with $z_j^0\in {\mathbb{S}}^1$ defining the base point $X_{{\mathbf{z}}^0}$ of the space $\mathbf{T}(0,n)=\mathbf{T}\left(X_{{\mathbf{z}}^0}\right)$. Its points are the equivalence classes $\left[\mu \right]$ of Beltrami coefficients from the ball $Belt{\left(\mathbb{C}\right)}_1=\{\mu \in L_{\infty }{\left(\mathbb{C}\right):\parallel \mu \parallel }_{\infty }<1\}$ under the relation: ${\mu }_1\sim {\mu }_2$, if the corresponding quasiconformal homeomorphisms $w^{{\mu }_1},w^{{\mu }_2}:X_{{\mathbf{a}}^0}\to X_{\mathbf{z}}$ are homotopic on $X_{{\mathbf{z}}^0}$ (and hence coincide in the points $0,1,z_1^0,\dots ,z_{n-3}^0,\infty$). This models $\mathbf{T}(0,n)$ as the quotient space $\mathbf{T}(0,n)=Belt{\left(\mathbb{C}\right)}_1/\sim$ with complex Banach structure of dimension $n-3$ inherited from the ball $Belt{\left(\mathbb{C}\right)}_1$.

Another canonical model of the space $\mathbf{T}(0,n)=\mathbf{T}\left(X_{{\mathbf{z}}^0}\right)$ is obtained again using the uniformization. The surface $X_{{\mathbf{z}}^0}$ is conformally equivalent to the quotient space $U/{\Gamma }_0$, where ${\Gamma }_0$ is a torsion-free Fuchsian group of the first kind acting discontinuously on $\mathbb{D}\cup {\mathbb{D}}^{*}$. The functions $\mu \in L_{\infty }\left(X_{{\mathbf{z}}^0}\right)$ are lifted to $\mathbb{D}$ as the Beltrami $(-1,1)$-measurable forms $\tilde{\mu }d\overline{z}/dz$ in $\mathbb{D}$ with respect to ${\Gamma }_0$ which satisfy $(\tilde{\mu }\circ \gamma )\overline{{\gamma }^{\prime }}/{\gamma }^{\prime }=\tilde{\mu },\gamma \in {\Gamma }_0$, and form the Banach space $L_{\infty }(\mathbb{D},{\Gamma }_0)$. After extending these $\tilde{\mu }$ by zero to ${\mathbb{D}}^{*}$, the Schwarzians $S_{w^{\tilde{\mu }}|{\mathbb{D}}^{*}}$ for $\parallel \tilde{\mu }{\parallel }_{\infty }<1$ belong to $\mathbf{T}$ and form its subspace regarded as the Teichmüller space $\mathbf{T}\left({\Gamma }_0\right)$ of the group ${\Gamma }_0$ of holomorphic ${\Gamma }_0$-automorphic forms of degree $-4$. This represents the space $\mathbf{T}\left(X_{{\mathbf{z}}^0}\right)$ as a bounded domain in the complex Euclidean space ${\mathbb{C}}^{n-3}$.

Any Teichmüller space is a complete metric space 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., [15,18,19].

3. Proof of Theorem 1

We first establish that the equalities (7) imply the extremality of ${\kappa }_{\theta }$. This will be given in three steps following the lines of [4,5].

Step 1: Renormalization of functions and lifting the coefficient functionals onto spaces $\mathbf{T}$ and ${\mathbf{T}}_1$.

Lemma 1 allows us to involve more general classes $S_{Q,\theta }\left(\mathbb{D}\right)$ of univalent functions in the disk $\mathbb{D}$ with expansions

$$f\left(z\right)=e^{i\theta }z+a_2{z}^2+\dots ,-\pi \le \theta \le \pi ,$$

admitting quasiconformal extension to ${\mathbb{D}}^{*}$, and their subclasses $S_{z_0,\theta }$ consisting of $f\in S_{Q,\theta }$ with fix point at $z_0\in {\mathbb{S}}^1$. The corresponding classes of univalent functions

$$F\left(z\right)=e^{-i\theta }z+b_0+b_1{z}^{-1}+b_2{z}^{-2}+\dots .$$

are denoted by ${\Sigma }_{Q,\theta }$ and ${\Sigma }_{z_0,\theta }$. Consider their disjunct unions

$$S^0=\bigcup _{z_0\in {\mathbb{S}}^1,\theta \in [-\pi ,\pi ]}{S}_{z_0,\theta },{\Sigma }^0=\bigcup _{z_0\in {\mathbb{S}}^1,\theta \in [-\pi ,\pi ]}{\Sigma }_{z_0,\theta },$$

and note that their closures $\overline{S^0},\overline{{\Sigma }^0}$ in the topology of locally uniform convergence on the sphere $\widehat{\mathbb{C}}$ are compact. The given functional $J\left(f\right)$ naturally extends to these generalized classes.

Similar to (4), the coefficients $a_n$ of $f\left(z\right)\in S_{Q,\theta }$ and the corresponding coefficients $b_j$ of inversions $F_f\left(z\right)1/f(1/z)$ are related by

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

(11)

where ${ϵ}_{n,j}$ are the entire powers of $e^{i\theta }$. This again transforms the initial functional $J\left(f^{\mu }\right)$ on $S^0$ into a coefficient functional $\tilde{J}\left(F^{\mu }\right)$ on ${\Sigma }^0$ depending on the corresponding coefficients $b_j$. This dependence is holomorphic from the Beltrami coefficients ${\mu }_F\in Belt{\left(\mathbb{D}\right)}_1$ and from the Schwarzians $S_{F^{\mu }}$ and generates holomorphic lifting the original functionals $J\left(f\right)$ and $\tilde{J}\left(F\right)$ onto the universal Teichmüller space $\mathbf{T}\subset \mathbf{B}$ as holomorphic functions of $S_F\in \mathbf{T}$.

Our next goal is to lift J onto the covering space ${\mathbf{T}}_1$. To reach this, we pass again to the functional $\widehat{J}\left(\mu \right)=\tilde{J}\left(F^{\mu }\right)$. This relation lifts J onto the ball $Belt{\left(\mathbb{D}\right)}_1$.

Now we apply the ${\mathbf{T}}_1$-equivalence of maps $f^{\mu }$, i.e., the quotient map

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

which involves the homotopy of maps $F^{\mu }$ on the punctured disk $\mathbb{D}\setminus \left\{0\right\}$. Thereby the functional $\tilde{J}\left(F^{\mu }\right)$ is pushed down to a bounded holomorphic functional $\mathcal{J}\left(X_{F^{\mu }}\right)$ on the space ${\mathbf{T}}_1$. We denote this functional by $\mathcal{J}$.

The Bers isomorphism theorem allows one to regard the points of the space ${\mathbf{T}}_1=Fib\left(\mathbf{T}\right)$ as the pairs $X_{F^{\mu }}=(S_{F^{\mu }},F^{\mu }\left(0\right))$, where $\mu \in Belt{\left(\mathbb{D}\right)}_1$ obey ${\mathbf{T}}_1$-equivalence, which implies a logarithmically plurisubharmonic functional

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

(12)

defined on the whole space $FibF\left(\mathbf{T}\right)$.

Step 2: Subharmonicity of maximal function generated by $\mathit{J}$. The functional (12) generates for any fixed $\theta \in [-\pi ,\pi ]$ and $F^{\mu }\in {\Sigma }_{Q,\theta }$ the maximal function

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

(13)

on the range domain $D_{\theta }$ of $F^{\mu }\left(0\right)$, taking the supremum over all $S_{F^{\mu }}\in \mathbf{T}$ admissible for a given $t=F^{\mu }\left(0\right)\in D_{\alpha }$ (that means over the pairs $(S_{F^{\mu }},t)\in Fib\left(\mathbf{T}\right)$ with a fixed t).

The crucial step in the proof of Theorem 1 is to establish that every function (13) inherits from ${\mathcal{Z}}_n$ subharmonicity in t, which we present as

Lemma 2.

Every function $u_{\theta }\left(t\right)$ with a fixed $\theta \in [-\pi ,\pi ]$ is logarithmically subharmonic in some domains $D_{\theta }$ located in the disk ${\mathbb{D}}_4=\left\{\right|t|<4\}$.

Proof. 

Fix $\theta \in [-\pi ,\pi ]$ and, using the maps $F^{\mu }\in {\Sigma }_{Q,\theta }$, 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 the unit circle) 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 ${\Gamma }_m$ of covering transformations uniformizing $X_m^{\prime }$, which act discontinuosly in both disks $\mathbb{D}$ and ${\mathbb{D}}^{*}$.

Every such group $G_m$ has a canonical (open) fundamental polygon $P_m$ of ${\Gamma }_m$ in $\mathbb{D}$ corresponding to the dissection $L_m$. It is a regular circular $2^{n+1}$-gon centered at the origin of the disk and can be chosen to have a vertex at the point $z=1$. The restriction of $g_m$ to $P_m$ is univalent, and as $m\to \infty$, these polygons entirely increase and exhaust the disk $\mathbb{D}$.

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 $F^{\mu }\in {\Sigma }_{Q,\theta }$ 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 (where $k={\parallel \mu \parallel }_{\infty }$) and compatible with the group ${\Gamma }_m$. 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 $L_{\infty }$; hence, $F^{{\mu }_m}\left(0\right)$ is a holomorphic function of $t=F^{\mu }\left(0\right)$.

As a result, one obtains that the Beltrami coefficients

$${\mu }_{h,m}:={\left[g_m\right]}_{*}{\mu }_h$$

and the corresponding values $F^{{\mu }_{h,m}}\left(0\right)$ are holomorphic functions of the variable $t=F^{\mu }\left(0\right)$.

By Hartogs theorem, the function $\mathcal{J}(S_{F^{{\mu }_m}},t)$ with $t=F^{{\mu }_m}\left(0\right)$ is jointly holomorphic in $(S_{F^{{\mu }_m}},t)\in \mathcal{F}\left(\mathbf{T}\right)$.

We now choose in $\mathbf{T}(0,m)\setminus \left\{\mathbf{0}\right\}$ represented as a subdomain of the space $\mathbf{B}\left({\Gamma }_m\right)$ a countable dense subset

$$E^{\left(m\right)}=\{{\phi }_1,{\phi }_2,\dots ,{\phi }_p,\dots \}.$$

For any of its point ${\phi }_p$, the corresponding extremal Teichüller disk $\mathbb{D}\left({\phi }_p\right)$ joining this point with the origin of $\mathbf{B}\left({\Gamma }_m\right)$ does not meet other points from this set (this follows from the uniqueness of Teichmüller extremal map). Recall also that each disk $\mathbb{D}\left({\phi }_p\right)$ is formed by the Schwarzians $S_{F^{\tau {\mu }_{p;m}}}$ with $\left|\tau \right|<1$ and

$${\mu }_{p;m}\left(z\right)=\left|{\psi }_{p;m}\left(z\right)\right|/{\psi }_{p;m}\left(z\right)$$

with appropriate ${\psi }_{p;m}\in A_1(\mathbb{D},{\Gamma }_m),{\parallel {\psi }_{p;m}\parallel }_1=1$.

The restrictions of the functional $\mathcal{J}(S_{F^{\tau {\mu }_{p;m}}},t)$ to these disks are holomorphic functions of $(\tau ,t)$; moreover, the above construction provides that all these restrictions are holomorphic in t in some common domain $D_m\subset {\mathbb{D}}_4$ containing the point $t=0$, provided that $\left|\tau \right|\le k<1$. We use the maximal common holomorphy domain; it is located in a disk $\left\{\right|t|<r_0\},r_0<4$.

Maximization over $\tau$ implies the logarithmically subharmonic functions

$$U_{p;m}\left(t\right)=\underset{\left|\tau \right|<1}{sup}\left|\mathcal{J}(S_{F^{\tau {\mu }_{p;m}}},t)\right|(t=F^{{\mu }_{p;m}}\left(0\right),p=1,2,\dots )$$

in the domain $D_m$. We consider the upper envelope of this sequence

$$u_m\left(t\right)=\underset{p}{sup}{U}_{p;m}\left(t\right)$$

defined in some domain $D_m\subset {\mathbb{D}}_4$ containing the origin, and take its upper semicontinuous regularization

$$u_m\left(t\right)=\underset{t^{\prime }\to t}{lim\; sup}{u}_m\left(t^{\prime }\right),$$

which does not increase $max\left|\mathcal{J}\right|$ (by abuse of notation, we shall denote the regularizations by the same letter as the original functions).

Repeating this for all m, one obtains the sequences of monotone increasing functions $u_m\left(t\right)$ and of increasing domains $D_m$ exhausting a domain $D_{\theta }={\bigcup }_m{D}_m$ such that each $u_m$ is subharmonic on $D_m$, and the limit function of this sequence is equal to the function (13). It is defined and subharmonic on the domain $D_{\theta }$. The lemma follows. □

Step 3: Majorization on cover of ${\Sigma }_{\mathit{af}}$ and Koebe’s function. The above construction leads to upper semicontinuous envelope $u\left(t\right)={sup}_{\theta }{u}_{\theta }\left(t\right)$ of functions (13) that are logarithmically subharmonic in some domain $D_0$ (which, in view of weak rotational homogeneity of J, is a disk ${\mathbb{D}}_a=\left\{\right|t|<a\}$ of some radius $a\le 4$, and generically ${max}_{t}u\left(t\right)>{max}_S\left|J\left(f\right)\right|$.

We now show that restricting the functional $\mathcal{J}$ to the image of ${\Sigma }_{af}$ in ${\mathbf{T}}_1$, one obtains the best upper subharmonic dominant for $\mathcal{J}$, which intrinsically relates to ${\kappa }_{\theta }$.

First, we establish the properties of the image of ${\Sigma }_{af}$ in the underlying space $\mathbf{T}$. Denote this image by $G_{af}$. Its structure is described by the following

Lemma 3

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

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 $w\left(z\right)=e^{i\theta }z+{\sum }_2^{\infty }{a}_n{z}^n$). Put

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

(14)

and let $a_{2,\theta }^0\ne 0$ and $w_0\left(z\right)=e^{i\theta }z+a_2^0{z}^2+\dots$ be one of the maximizing functions. 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 $\left\{\right|W+a_{2,\theta }^0|\le |a_{2,\theta }^0\left|\right\}$.

This lemma implies that the image of the set ${\Sigma }_{af}$ is a three-dimensional subdomain in the space $\mathbf{T}$ and its image in $Fib\left(\mathbf{T}\right)$ is a complex four-dimensional subdomanifold of $Fib\left(\mathbf{T}\right)$.

We denote these images by ${\Sigma }_{af}$ and $Fib\left({\Sigma }_{af}\right)$, respectively, and take the restriction of functional $\left|\mathcal{J}\right(S_{F^{\mu }},t\left)\right|$ onto the second set. Our goal now is to maximize this restricted functional.

We select a dense subsequence $\{{\theta }_1,{\theta }_2,\dots \}\subset [-\pi ,\pi ]$ and define the corresponding functionals on the classes $S_{{\theta }_j}$ and ${\Sigma }_{{\theta }_j}$ replacing the original functional $\tilde{J}$ as follows. Having the functions

$$f_a\left(z\right)=e^{i{\theta }_m}z+a_2{z}^2+\dots$$

and the corresponding

$$F_a\left(z\right)=1/f_a\left(z\right)=e^{-i{\theta }_m}z+b_0+b_1{z}^{-1}+\dots ,a=e^{i{\theta }_m},$$

consider $z^{\prime }=e^{i{\theta }_m}z$ as a new independent variable. Then

$$f_a\left(z^{\prime }\right)=z^{\prime }+e^{-2i{\theta }_m}{\left(z^{\prime }\right)}^2+\dots ,F_a\left(z^{\prime }\right)=z^{\prime }+b_0{e}^{i{\theta }_m}+b_1{e}^{i{\theta }_m}{\left(z^{\prime }\right)}^{-1}+\dots$$

belong to $S_Q$ and ${\Sigma }_Q$ (in terms of variable $z^{\prime }$).

Noting that $J\left(f\right)$ is a polynomial of the form

$$J\left(f\right)=\sum _{\left|\alpha \right|=n_0}^N{C}_{m_1,\dots ,m_s}{a}_{m_1}^{{\alpha }_{m_1}}\dots a_{m_s}^{{\alpha }_{m_s}}$$

with $\left|\alpha \right|={\alpha }_{m_1}+\cdots +{\alpha }_{m_s}\ge 3$, we set

$${\tilde{J}}_{{\theta }_m}\left(F_a\right)=e^{2i{m}_1\theta }{a}_{m_1}^{{\alpha }_{m_1}}\dots e^{2i{m}_s\theta }{a}_{m_s}^{{\alpha }_{m_s}}.$$

(15)

Therefore,

$$\underset{{\Sigma }_{{\theta }_j}}{max}|{\tilde{a}}_{m_j}^{{\alpha }_{m_j}}|=\underset{\Sigma }{max}|\tilde{J}\left(F_f\right)|=\underset{S}{max}\left|J\left(f\right)\right|,$$

(16)

and similarly for the corresponding collections ${\Sigma }_{af,{\theta }_j}$ of functions

$$F_{b_0,b_1;t}^{\theta }\left(z\right)=e^{i\theta }z+b_{0}t+b_1{t}^2{z}^{-1}.$$

Now consider the sequence of increasing products of the quotient spaces

$${\mathcal{T}}_m=\prod _{j=1}^m{\widehat{\Sigma }}_{{\theta }_j}/\sim =\prod _{j=1}^m\left\{(S_{F_{{\theta }_j}},F_{{\theta }_j}^{{\mu }_j}\left(0\right))\right\}\simeq {\mathbf{T}}_1\times \cdots \times {\mathbf{T}}_1,$$

(17)

where the equivalence relation ∼ again means ${\mathbf{T}}_1$-equivalence. The Beltrami coefficients ${\mu }_j\in Belt{\left(\mathbb{D}\right)}_1$ are chosen here independently. For any ${\mathbf{T}}_1$, presented in the right-hand side of (17), the corresponding values of $F_{{\theta }_j}^{{\mu }_j}\left(0\right)$ run over some domain $D_{{\alpha }_j}\subset \mathbb{C}$, and the corresponding collection $\beta =({\beta }_1,\dots ,{\beta }_s)$ of the Bers isomorphisms

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

determines a holomorphic surjection of the space ${\mathcal{T}}_m$ onto the product of m spaces $\mathcal{F}\left(\mathbf{T}\right)$.

Letting

$${\mathbf{F}}_{{\theta }^{\mu }}\left(0\right):=(F_{{\theta }_1}^{{\mu }_1}\left(0\right),\dots ,F_{{\theta }_m}^{{\mu }_m}\left(0\right)),{\mathbf{S}}_{{\mathbf{F}}_{{\theta }^{\mu }}}:=(S_{F_{{\theta }_1}},\dots ,S_{F_{{\theta }_m}}),$$

consider the holomorphic maps (vector-functions)

$$\mathbf{h}\left({\mathbf{S}}_{{\mathbf{F}}_{\theta }}\right)=(h_1\left(S_{F_{{\theta }_1}}\right),\dots h_m\left(S_{F_{{\theta }_m}}\right)):{\widehat{\Sigma }}_{af,\theta }:={\Sigma }_{af,{\theta }_1}\times \cdots \times {\Sigma }_{af,{\theta }_m}\to {\mathbb{C}}^m,m=1,2,\dots ,$$

with

$$h_j\left(S_{F_{{\theta }_j}}\right)={\tilde{Z}}_{n,{\theta }_j}\left(F_a\right),j=1,\dots ,m,$$

endowed with the polydisk norm

$$\parallel \mathbf{h}\parallel =\underset{j}{max}\left|h_j\right|$$

on ${\mathbb{C}}^m$. Then by (16),

$$\underset{{\Sigma }_{af}}{max}\parallel \mathbf{h}\parallel =\underset{j}{max}\underset{{\Sigma }_{af}}{max}|h_j(S_{F_{{\theta }_j}})|=\underset{S}{max}\left|Z\left(f\right)\right|.$$

(18)

The image of the set ${\Sigma }_{af}$ under this embedding is the set ${Fib}_m\left(G_{af}\right)$, the free product of m factors ${Fib}_j\left(G_{af}\right)$. Note that its dimension equals $4m$ and that restriction of $\mathbf{h}$ to ${\Sigma }_{af}$ is a polynomial map.

We now apply the construction from the previous step simultaneously to each component $h(S_{F_{{\theta }_j}},t)$ on the corresponding space ${\mathbf{T}}_1$ in (16) and obtain in the same fashion that the function

$$u_m\left(t\right)=max\left(|h\left(S_{F_{{\theta }_1}}\right),t\left)\right|,\dots ,|h\left(S_{F_{{\theta }_m}}\right),t\left)\right|\right)$$

is subharmonic in some disk ${\mathbb{D}}_{a_m}$ of radius $a_m\le 4$.

The rotational symmetry of the joint domain ${\mathbb{D}}_m$ follows from the symmetry of the set ${\Sigma }_{af}$ and of its image in ${\mathcal{T}}_m$ and from the following variational lemma, which is a special case of the general quasiconformal deformations constructed in [3].

Lemma 4.

Let D be a simply connected domain on the Riemann sphere $\widehat{\mathbb{C}}$. Assume that there is a set E of positive two-dimensional Lebesgue measure and a finite number of points $z_1,z_2,\dots ,z_m$ distinguished in D. Let ${\alpha }_1,{\alpha }_2,\dots ,{\alpha }_m$ be non-negative integers assigned to $z_1,z_2,\dots ,z_m$, respectively, so that ${\alpha }_j=0$ if $z_j\in E$.

Then, for a sufficiently small ${\epsilon }_0>0$ and $\epsilon \in (0,{\epsilon }_0)$, and for any given collection of numbers $w_{sj},s=0,1,\dots ,{\alpha }_j,j=1,2,\dots ,m$ which satisfy the conditions $w_{0j}\in D$,

$$|w_{0j}-z_j|\le \epsilon ,|w_{1j}-1|\le \epsilon ,|w_{sj}|\le \epsilon (s=0,1,\dots a_j,j=1,\dots ,m),$$

there exists a quasiconformal automorphism h of D which is conformal on $D\setminus E$ and satisfies

$$h^{\left(s\right)}\left(z_j\right)=w_{sj}\mathit{for}\mathit{all}s=0,1,\dots ,{\alpha }_j,j=1,\dots ,m.$$

Moreover, the Beltrami coefficient ${\mu }_h\left(z\right)={\partial }_{\overline{z}}h/{\partial }_{z}h$ of h on E satisfies $\parallel {\mu }_h{\parallel }_{\infty }\le M\epsilon$. The constants ${\epsilon }_0$ and M depend only upon the sets $D,E$ and the vectors $(z_1,\dots ,z_m)$ and $({\alpha }_1,\dots ,{\alpha }_m)$.

If the boundary $\partial D$ is Jordan or is $C^{l+\alpha }$-smooth, where $0<\alpha <1$ and $l\ge 1$, we can also take $z_j\in \partial D$ with ${\alpha }_j=0$ or ${\alpha }_j\le l$, respectively.

We apply this lemma to functions $F\in {\Sigma }_Q$ and take the prescribed set E in domain $F\left(\mathbb{D}\right)$ to vary $F\left(0\right)$.

Each function $u_m\left(t\right)$ is a circularly symmetric function on its disk ${\mathbb{D}}_{a_m}$, and so is their upper envelope

$$u_J\left(t\right)=\underset{m\to \infty }{lim\; sup}{\tilde{u}}_m\left(t\right)$$

(on some disk ${\mathbb{D}}_a,a>0$). This envelope satisfies

$$\underset{{\mathbb{D}}_a}{max}{u}_J\left(t\right)=\underset{S}{max}\left|J\left(f\right)\right|$$

and attains its maximal value at the boundary point $t=a$.

Noting that the closure of ${\Sigma }_{af}$ contains the functions

$$F_{\theta }\left(z\right)=z-2e^{i\theta }+e^{2i\theta }{z}^{-1}$$

inverting the Koebe functions ${\kappa }_{\theta }$, one derives that the radius a must equal 4, which means that the range domain of $F^{\mu }\left(0\right)$ for $F^{\mu }\in {\Sigma }_{af}$ coincides with the disk ${\mathbb{D}}_4$. This yields that the boundary points of this domain correspond only to functions $f\left(z\right)$ with $|a_2|=2$, hence only to ${\kappa }_{\theta }\left(z\right)$.

By (7), this function is extremal also for the values of $J\left(f\right)$ on the whole class S, which completes the proof of the part “if”.

Step 4: Part “only if”. It remains to establish that extremality of ${\kappa }_{\theta }$ on S provides the relations (7).

Passing, if needed, to the normalized functional

$$J^0\left(f\right)=J\left(f\right)/\underset{S}{max}\left|J\left(f\right)\right|,$$

one can assume that the coefficients $C_{m_1,\dots ,m_s}$ of $J\left(f\right)$ are such that $\left|J\right(f\left)\right|\le 1$ on S. Then

$$J\left({\kappa }_{\theta }\right)=\sum _{\left|\alpha \right|=n_0}^N{C}_{m_1,\dots ,m_s}{m}_1^{{\alpha }_{m_1}}{e}^{i{m}_1\theta }\dots m_s^{{\alpha }_{m_s}}{e}^{i{m}_s\theta }=e^{i\sigma \left(\theta \right)}$$

(since $\left|J\right({\kappa }_{\theta }\left)\right|=1$), and for all $\left|t\right|<1$,

$$J({\kappa }_{\theta },t)=\sum _{\left|\alpha \right|=n_0}^N{C}_{m_1,\dots ,m_s}{m}_1^{{\alpha }_{m_1}}{t}^{m_1}{e}^{i{m}_1\theta }\dots m_s^{{\alpha }_{m_s}}{t}^{m_s}{e}^{i{m}_s\theta }=e^{i\sigma \left(\theta \right)}t.$$

Thus, the differential metric ${\lambda }_J$ defined by the holomorphic map $J({\kappa }_{\theta },t):\mathbb{D}\to \mathbb{D}$ on the unit disk (and simultaneously the homotopy and extremal disks of ${\kappa }_{\theta }$ in the space $\mathbf{T}$) is connected with the hyperbolic metric of the unit disk by

$${\lambda }_J\left(t\right)={\lambda }_{\mathbb{D}}\left(t^{1/p}\right)={\displaystyle \frac{{p\left|t\right|}^{p-1}}{1-{\left|t\right|}^{2p}}},t\in \mathbb{D}.$$

That implies that the conformal metrics determined by all holomorphic maps $h:\mathbb{D}\to \mathbf{T}$ (via pull backing of ${\lambda }_{\mathbb{D}}$) generated by functions $h\left(t\right)=f_t\left(z\right),f\in S$, must satisfy on their holomorphic disks $h\left(\mathbb{D}\right)\subset \mathbf{T}$ and on $\mathbb{D}$ the inequality

$${\lambda }_h\left(t\right)\le {\lambda }_J\left(t\right)\mathrm{for}\mathrm{any}t\in \mathbb{D},$$

This is a consequence of the following lemma, which is a straightforward extension of the classical Ahlfors-Schwarz lemma.

Lemma 5.

Let $\lambda \left(t\right)\left|dt\right|$ be a continuous conformal metric on the disk $\mathbb{D}$ with growth

$$\lambda \left(\right|t\left|\right)={pc\left|t\right|}^{p-1}+{O\left(\right|t|}^p\left)\mathrm{as}\right|t|\to 0\mathrm{with}0<c\le 1(p=1,2,\dots )$$

near the origin of Gaussian curvature ${\kappa }_{\lambda }\le -4$ in the supporting sense at its noncritical points. Then

$$\lambda \left(t\right)\le {p\left|t\right|}^{p-1}{/(1-|t|}^{2p})\mathit{for}\mathit{all}t\in \mathbb{D}.$$

The curvature is defined by

$${\kappa }_{\lambda }=-\Delta log\lambda /{\lambda }^2$$

and can be understood here even in the generalized sense, i.e., with the distributional Laplacian $\Delta =4\partial \overline{\partial }$.

Accordingly, the corresponding integral distances on $\mathbb{D}$ generated by metrics ${\lambda }_J$ and ${\lambda }_h$ must satisfy for any $f\in S$ the inequality

$$\left|J\left(f\left(tz\right)\right)\right|\le |J\left({\kappa }_{\theta }\left(tz\right)\right)|=|t\left|\mathrm{for}\mathrm{any}\right|t|\le 1,$$

where the equality (even on one pair $(t,z)$) is valid only if $f\left(z\right)={\kappa }_{\theta }\left(z\right)$.

Since the homotopy (and simultaneously extremal) disk of the function ${\kappa }_{\theta }$ is located entirely in the set ${\Sigma }_{af}$, this implies the equalities (7), completing the proof of Theorem 1.