A General Coefficient Theorem for Univalent Functions: Generalization of the Bieberbach and Zalcman Conjectures

Abstract

The main result of this paper is that any rotationally homogeneous polynomial functional $J\left(f\right)={\displaystyle \sum _{\left|\alpha \right|=n_0}^N}{C}_{m_1,\dots ,m_s}{a}_{m_1}^{{\alpha }_{m_1}}\dots a_{m_s}^{{\alpha }_{m_s}}$ with $\left|\alpha \right|={\alpha }_{m_1}+\cdots +{\alpha }_{m_s}\ge 3,2<a_{m_1}<\cdots <a_{m_s}<\infty$, depending on the distinguished finitely many coefficients $a_{m_j}$, is maximized on S by Koebe’s function ${\kappa }_{\theta }\left(z\right)=z/{(1-e^{i\theta }z)}^2$ with $\theta \in [-\pi ,\pi ]$. This includes, in particular, the well-known Bieberbach and Zalcman conjectures and covers many other coefficient estimates for univalent functions. As an application, the main theorem provides the solution of the generalized Zalcman conjecture posed by Ma.

1. Introduction

1.1. Statement of Problem

Sharp estimating holomorphic functionals on various classes of univalent functions depending on the Taylor coefficients of these functions has classical origins but still remains a very complicated important problem in geometric complex analysis actively investigated by many authors (see, e.g., the books [1,2,3,4,5,6,7,8,9] and the references cited there). Such functionals play a significant role in various geometric and physical applications of complex analysis.

Among the brilliant conjectures in geometric function theory, there were the Bieberbach and Zalcman conjectures investigated by many authors and remained a long time open.

Recently the author established in [10] that these conjectures are equivalent and gave their simultaneous proof. It turns out that in this way one can solve a much more general problem generalizing both conjectures.

We consider the canonical class S of univalent functions on the unit disk $\mathbb{D}=\left\{\right|z|<1\}$ with expansions $f\left(z\right)=z+{\displaystyle \sum _2^{\infty }}{a}_n{z}^n$ (i.e., with $f\left(0\right)=0,f^{\prime }\left(0\right)=1$) and are concerned with the following problem, generalizing the indicated conjectures:

Does any weakly rotationally homogeneous polynomial functional

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

(1)

with

$$\left|\alpha \right|={\alpha }_{m_1}+\cdots +{\alpha }_{m_s}\ge 3,2<a_{m_1}<\cdots <a_{m_s}<\infty ,$$

depending on the distinguished finitely many coefficients $a_{m_j}$, be maximized on any subclass of S containing the Koebe function

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

(2)

by this function?

The strong rotational homogeneity means the invariance

$$J\left(f_{\alpha ,\beta }\right)=J\left(f\right)e^{i(\alpha +\beta )p}\mathrm{with}\mathrm{some}p\in \mathbb{N}$$

under arbitrary pre and post rotations

$$f\left(z\right)\mapsto f_{\alpha ,\beta }\left(z\right)=e^{i\beta }f\left(e^{i\alpha }z\right),$$

with independent $\alpha$ and $\beta$ from $[-\pi ,\pi ]$, while the weak homogeneity means that such equality holds only for $\beta =-\alpha$.

The results for strongly homogeneous functionals J established in [11,12] embrace much more general collections $\mathcal{X}$ and imply that ${max}_{\mathcal{X}}\left|J\left(f\right)\right|$ is attained on functions maximizing the second coefficient $a_2$ on $\mathcal{X}$; moreover, these results extend the general holomorphic functionals.

The weak homogeneity is more natural in the topics of geometric function theory dealing with the normalized collections of univalent functions, and many functionals obey only such homogeneity (for example, Zalcman’s functional $Z_n\left(f\right)=a_n^2-a_{2n-1},n\ge 3$).

The collection of strongly homogeneous functionals is sparse, and Theorem 1 increases widely the set of homogeneous functionals maximized by the Koebe function.

1.2. Main General Theorem

The following general theorem covers many quantitative results on coefficients and provides the proof of several well-known conjectures as the special cases. Also it illustrates the remarkable role of Koebe’s function.

Theorem 1.

Every weakly rotationally homogeneous polynomial functional (1), whose zero set ${\mathcal{Z}}_J=\{w\in \widehat{S}:J\left(w\right)=0\}$ is separated from the rotation set $\mathcal{K}=\{{\kappa }_{\tau ,\theta }\left(z\right)=e^{-i\theta }{\kappa }_0\left(e^{i\tau }z\right)\}$ of the Koebe function ${\kappa }_{\theta }$, is maximized on any rotationally invariant subclass $\mathcal{X}\subset S$ containing ${\kappa }_{\theta }$ only by this function and its rotations $e^{-i\alpha }{\kappa }_{\theta }\left(e^{i\alpha }z\right)$.

So, unless $J\left({\kappa }_{\theta }\right)=0$, only this function is extremal for any weakly rotationally homogeneous coefficient functional J on S and, thereby, on the rotationally invariant subfamilies $\mathcal{X}\subset S$ containing ${\kappa }_{\theta }$.

The proof is obtained by the same approach as in [11,12], which involves some deep analytic and geometric results from Teichmüller space theory, especially the Bers isomorphism theorem. The functional J is lifted to the Teichmüller space ${\mathbf{T}}_1$ of the punctured disk ${\mathbb{D}}_{*}=\{0<|z|<1\}$. This space is biholomorphically equivalent to the Bers fiber space $\mathcal{F}\left(\mathbf{T}\right)$ over the universal Teichmüller space $\mathbf{T}=Teich\left(\mathbb{D}\right)$. This generates a holomorphic functional $\mathcal{J}(\phi ,t)$ on $\mathcal{F}\left(\mathbf{T}\right)$ covering J, with the same range domain. Here $\phi =S_F\in \mathbf{T}$ are the Schwarzian derivatives of functions $F\in {\mathrm{\Sigma }}_{\theta }\left(1\right)$, while the variable t runs over the fiber domain $F_{\phi }\left(\mathbb{D}\right)$ defined by $\phi$.

A crucial step is to maximize $\left|\mathcal{J}\right(\phi ,t\left)\right|$ over $\phi$ by a fixed t. We apply an approximation of the underlying space $\mathbf{T}$ by the finitely dimensional Teichmüller spaces of the punctured spheres in the weak topology of locally uniform convergence in $\mathbb{C}$. Any such space is foliated by Teichmüller-Kobayashi geodesic disks. We deal with restrictions of $\mathcal{J}(\phi ,t)$ to these disks, taking their appropriate dense countable collection. This implies a maximal logarithmically subharmonic function $u_{\theta }\left(t\right)>0$ on a domain located in the disk ${\mathbb{D}}_4=\left\{\right|t|<4\}$.

Repeating this construction for all $\theta$, one creates a logarithmically subharmonic function $u\left(t\right)=sup{u}_{\theta }\left(t\right)$ on the disk ${\mathbb{D}}_4$ with

$$\underset{t}{max}u\left(t\right)=\underset{{\mathbf{T}}_1}{sup}\left|\mathcal{J}\right|=\underset{S}{max}\left|J\right|.$$

This maximal value is attained on the boundary of ${\mathbb{D}}_4$ whose points correspond to the function ${\kappa }_0\left(z\right)$ composed with rotations.

A deep open question is to describe the extremals of weakly homogeneous functionals $J\left(f\right)$ on subclasses of S not containing Koebe’s function.

1.3. Some Associated Quantities

Note that all functions $f_{\alpha ,\beta }\left(z\right)$ have the same Schwarzian derivative

$$S_f\left(z\right)={\left({\displaystyle \frac{f^{″}\left(z\right)}{f^{\prime }\left(z\right)}}\right)}^{\prime }-{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{f^{″}\left(z\right)}{f^{\prime }\left(z\right)}}\right)}^2(z\in \mathbb{D}),$$

and the chain rule

$$S_{f_1\circ f}\left(z\right)=(S_{f_1}\circ f)f^{\prime }{\left(z\right)}^2+S_f\left(z\right).$$

yields for the Möbius (fractional linear) maps $w=\gamma \left(z\right)$ of $\widehat{\mathbb{C}}$ the equalities

$$S_{f_1\circ \gamma }\left(z\right)=(S_{f_1}\circ \gamma ){\gamma }^{\prime }{\left(z\right)}^2,S_{\gamma \circ f}\left(z\right)=S_f\left(z\right).$$

Hence, each $S_f\left(z\right)$ can be regarded as a quadratic differential $\phi =S_f\left(z\right)d{z}^2$ on $\mathbb{D}$. The solution $w\left(z\right)$ of the Schwarzian equation $S_w\left(z\right)=\phi \left(z\right)$ with a given holomorphic $\phi$ is defined up to a Möbius transformation of $\widehat{\mathbb{C}}$.

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 bounded holomorphic functions on D with the norm

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

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, for the unit disk,

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

(see, e.g., [2,13]). The space $\mathbf{B}\left(D\right)$ is dual to the Bergman space $A_1\left(D\right)$, a subspace of $L_1\left(D\right)$ formed by integrable holomorphic functions (quadratic differentials $\phi \left(z\right)d{z}^2$) on D.

We shall also use the inverted functions

$$F_f\left(z\right)=1/f(1/z)=b_0+b_1{z}^{-1}+\cdots +b_n{z}^{-n}+\cdots ,\left|z\right|>1,$$

of functions $f\in S$, 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 $\mathrm{\Sigma }$, and 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 $Belt{\left({\mathbb{D}}^{*}\right)}_1$ and $Belt{\left(\mathbb{D}\right)}_1$ 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).

The Schwarzians $S_f$ of $f\in S_Q$ run over a bounded domain in $\mathbf{B}$ modeling the universal Teichmüller space $\mathbf{T}$ (and similarly $S_F$ of $F\in {\mathrm{\Sigma }}_Q$).

2. Some Applications of Theorem 1

Reformulation of Theorem 1

Theorem 1 can be reformulated in the following form:

Theorem 2.

For any weakly rotationally homogeneous polynomial functional (1) and any $f\in S$, we have the bound

$$\left|J\left(f\right)\right|\le \left|\sum _{\left|\alpha \right|=n_0}^N{C}_{m_1,\dots ,m_s}{m}_1^{{\alpha }_{m_1}}\dots m_s^{{\alpha }_{m_s}}\right|,$$

with equality only for the Koebe function ${\kappa }_{\theta }$.

This easily shows that Theorem 1 yields the proofs of several coefficients conjectures as the special cases. In particular, this includes the Bieberbach conjecture (with $J\left(f\right)=a_n$) and the Zalcman conjecture (with $J\left(f\right)=a_n^2-a_{2n-1}$), both investigated by many authors and remained open a long time.

Recently the author established in [10] that these conjectures are equivalent and gave their simultaneous proof. It turns out that in this way one also can solve more general problems. In particular, Theorem 1 provides the proof of the generalized Zalcman conjecture posed by Ma [14], which states:

For all $f\left(z\right)\in S$ and all $n\ge 2,m\ge 2$ must be

$$|a_n{a}_m-a_{n+m-1}|\le (n-1)(m-1),$$

(3)

with equality for the Koebe function (2) and its rotations.

This conjecture was proved in some very restricted cases (see [14,15,16]). Theorem 1 implies the complete result:

Theorem 3.

The estimate (3) holds for any function $f\left(z\right)\in S$, and the equality is valid only for the Koebe function ${\kappa }_{\theta }$.

3. Preliminary Results

3.1. A Distinguished Subclass of $\mathrm{\Sigma }$

For each $F\left(z\right)=z+b_0+b_1{z}^{-1}+\dots \in \mathrm{\Sigma }$, 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 ${\mathrm{\Sigma }}_Q$.

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 $k\le {\left|t\right|}^2$ (i.e., is satisfies on $\mathbb{D}$ the differential Beltrami equation ${\partial }_{\overline{z}}=\mu \left(z\right){\partial }_z$ with $\mu \in L_{\infty }\left(\mathbb{D}\right),{\parallel \mu \parallel }_{\infty }=k<1$).

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

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

(4)

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 [17], 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

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

where $F_{b_0,b_1;t}$ is the map (4) with $b_0,b_1$ coming from F. 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

$${\parallel \omega (·,t)\parallel }_{\mathbf{B}}=O\left(t^3\right),t\to 0;$$

(5)

this estimate is uniform for $\left|t\right|<t_0$; cf., e.g., [11,18].

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 ${\mathrm{\Sigma }}_Q$ (these values satisfy (5)). The collection of all such $F_t$ with $\left|t\right|<1$ will be denoted by ${\mathrm{\Sigma }}_{af}$. To have compactness, we shall also use the closure of this set with respect to locally uniform convergence on ${\mathbb{D}}^{*}$.

It suffices for our goals to consider the functions $f\in S$ with

$$b_1=a_2^2-a_3\ne 0;$$

this assumption is equivalent to $S_f\left(0\right)=\underset{z\to \infty }{lim}{z}^4{S}_{F_f}\left(z\right)\ne 0$. Such functions form a dense subset of S, and their Schwarzians form a dense subset of the space $\mathbf{T}$.

3.2. Restoration of Functional $J\left(f\right)$ by Its Infinitesimal Form

We pass to the normalized functional $J^0\left(f\right)=J\left(f\right)/{max}_S\left|J\left(f\right)\right|$ mapping S onto the unit disk and consider its action on infinite holomorphic families ${\mathcal{F}}_S=\{f_t\left(z\right)=f(z,t)\}\subset S$ and on ${\mathrm{\Sigma }}_{af}$ with $F(z,·)=1/f(1/z,·),t\in \mathbb{D}$.

Using the relations between the coefficients $a_n\left(t\right)$ of $f(z,t)$ and the corresponding coefficients $b_j\left(t\right)$ of $F_f(z,t)$, we represent $J^0$ as a polynomial functional on $\mathrm{\Sigma }$,

$$J^0\left(f\right)={\tilde{J}}^0\left(F_f\right)={\tilde{J}}^0(b_0,b_1,\dots ,,b_{2n-3}).$$

The given holomorphic families determine the sequences of holomorphic maps

$$g_m\left(t\right)=\tilde{J}\left(F_m(·,t)\right):\mathbb{D}\to \mathbb{C},m=1,2,\dots (F_m\in {\mathrm{\Sigma }}_{af}),$$

and their upper envelope

$$\widehat{g}=\underset{m}{sup}\left|g_m\left(t\right)\right|:\mathbb{D}\to \mathbb{D}$$

followed by upper semicontinuous regularization $\widehat{g}\left(t\right)=\underset{t^{\prime }\to t}{lim\; sup}{g}_m\left(t^{\prime }\right)$ presents a logarithmically subharmonic function on the unit disk.

The maps $g_m$ pull back the hyperbolic metric of this disk ${\lambda }_{\mathbb{D}}\left(z\right)\left|dz\right|$ generating on $\mathbb{D}$ the logarithmically subharmonic conformal metrics $ds={\lambda }_{g_m}\left(t\right)\left|dz\right|$ with

$${\lambda }_{g_m}\left(t\right)=g_m^{*}{\lambda }_{\mathbb{D}}\left(t\right)={\displaystyle \frac{|g_m^{\prime }\left(t\right)|}{1-|g_m{\left(t\right)|}^2}}$$

of Gaussian curvature $-4$ at noncritical points of $g_m$. Passing to the upper envelope

$${\lambda }_{J^0}\left(z\right)=\underset{m}{sup}{\lambda }_{g_m}\left(z\right)$$

(7)

report and its upper semicontinuous regularization, one obtains a logarithmically subharmonic metric on $\mathbb{D}$, whose curvature is less than or equal to $-4$ in both supporting and potential senses (cf. [10]).

We shall apply the results on the curvatures indicated above to the values of $\tilde{J}$ on the homotopy disks; thus, the derivatives must be understand as distributional, because generically these disks have the critical points.

The following restoration lemma provides that on extremal Teichmüller disks the functional $J^0$ can be reconstructed from its matric ${\lambda }_{J^0}$.

Lemma 1.

On any extremal Teichmüller disk $\mathbb{D}\left({\psi }_0\right)=\left\{t\right|{\psi }_0|/{\psi }_0:\left|t\right|<1\}\subset Belt{\left(\mathbb{D}\right)}_1$, we have the equality

$${tanh}^{-1}\left[{\tilde{J}}^0\left(F^{r|{\psi }_0|/{\psi }_0}\right)\right]=\underset{0}{\overset{r}{\mathit{\int }}}{\lambda }_{{\tilde{J}}^0}\left(t\right)dt$$

for each $r<1$.

The proof of this lemma is similar to the corresponding Lemma 7 in [10].

3.3. Special Quasiconformal Deformations

The following variational lemma ensures the existence of perturbations of maps $f^{\mu }$ whose domains $f^{\mu }\left(\mathbb{D}\right)$ have complements $\widehat{\mathbb{C}}\setminus f^{\mu }\left(\mathbb{D}\right)$ of positive area. It has many important applications (see, e.g., [10,11,12,13,19]).

Lemma 2

([13]). Let D be a simply connected domain on the Riemann sphere $\widehat{\mathbb{C}}$. Assume that there are 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.

4. A Glimpse to Teichmüleer 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 [20,21,22]. It is technically more convenient to deal with functions from ${\mathrm{\Sigma }}_Q$.

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 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)

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

letting ${\mu }_1,{\mu }_2\in Belt{\left(\mathbb{D}\right)}_1$ 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 [12] allows one to use some other normalizations of quasiconformally extendable functions.

Lemma 3.

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

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

(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 ${\mathbf{T}}_1=Teich\left({\mathbb{D}}_{*}\right)$ 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 $\mu \in Belt{\left(\mathbb{D}\right)}_1$ 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 $\mathsf{\Gamma }$, i.e., via $(\tilde{\mu }\circ \gamma )\overline{{\gamma }^{\prime }}/{\gamma }^{\prime }=\tilde{\mu },\gamma \in \mathsf{\Gamma }$, forming the Banach space $L_{\infty }(\mathbb{D},\mathsf{\Gamma })$; we extend these $\tilde{\mu }$ by zero to ${\mathbb{D}}^{*}$. Then ${\mathbf{T}}_1$ is canonically isomorphic to the subspace $\mathbf{T}\left(\mathsf{\Gamma }\right)=\mathbf{T}\cap \mathbf{B}\left(\mathsf{\Gamma }\right)$, where $\mathbf{B}\left(\mathsf{\Gamma }\right)$ consists of elements $\phi \in \mathbf{B}$ satisfying $(\phi \circ \gamma ){\left({\gamma }^{\prime }\right)}^2=\phi$ in ${\mathbb{D}}^{*}$ for all $\gamma \in \mathsf{\Gamma }$.

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

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

over the universal Teichmüller space $\mathbf{T}$ with holomorphic projection $\pi (\psi ,z)=\psi$ (see [20]).

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 $Fib\left(\mathbf{T}\right)$ 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 $z_j\in \mathbb{C}\setminus \{0,1\}$ (for details, see [10]).

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 [21,23,24]).

5. Proof of Theorem 1

We accomplish the proof in four steps. One can assume that $\mathcal{X}=S$.

Step 1: Renormalization of functions and lifting functional $J\left(f\right)$ onto spaces $\mathbf{T}$ and ${\mathbf{T}}_1$. Consider the 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 ${\mathrm{\Sigma }}_{Q,\theta }$ and ${\mathrm{\Sigma }}_{z_0,\theta }$. The closures $\overline{S^0},\overline{{\mathrm{\Sigma }}^0}$ of their disjunct unions

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

in the topology of locally uniform convergence on the sphere $\mathbb{C}$ are compact.

The family $\overline{S^0}$ closely relates to the class S, because every $w\in S$ has its representative $\widehat{w}$ in $\overline{S^0}$ (not necessarily unique) obtained by pre- and post-compositions of w with rotations $z\mapsto e^{i\alpha }z$ about the origin, related by $w_{\tau ,\theta }\left(z\right)=e^{-i\theta }w\left(e^{i\tau }z\right)$ with $\tau =arg{z}_0$, 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 $w\left(\mathbb{D}\right)$. The existence of such a point follows from the classical Schwarz lemma.

Now, using the relations between the coefficients $a_n$ of $f\left(z\right)\in S_{Q,\theta }$ and the corresponding coefficients $b_j$ of inversions $F_f\left(z\right)1/f(1/z)$, given by

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

where ${ϵ}_{n,j}$ are the entire powers of $e^{i\theta }$, one obtains successively the representations of $a_n$ by $b_j$:

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

(10)

These relations transform the initial functionals $J\left(f^{\mu }\right)$ into the coefficient functional $\tilde{J}\left(F^{\mu }\right)$ on ${\mathrm{\Sigma }}^0$ 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 $S_{F^{\mu }}$.

For any fixed $\theta$, the Taylor coefficients of functions $f^{\mu }\in S^0$ and $F^{\mu }\in {\mathrm{\Sigma }}^0$ depend holomorphically on $\mu \in Belt{\left(\mathbb{D}\right)}_1$ and on the Schwarzians $S_{F^{\mu }}$ as elements of $\mathbf{B}$. This 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 $\widehat{J}\left(\mu \right)=\tilde{J}\left(F^{\mu }\right)$ on the ball $Belt{\left(\mathbb{D}\right)}_1$ and apply the ${\mathbf{T}}_1$-equivalence of maps $f^{\mu }$, i.e., the quotient map

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

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

Now, using the Bers isomorphism theorem, we regard the points of the space ${\mathbf{T}}_1=Fib\left(\mathbf{T}\right)$ as the pairs $X_{F^{\mu }}=(S_{F^{\mu }},F^{\mu }\left(0\right))$ with $\mu \in Belt{\left(\mathbb{D}\right)}_1$ submitted to ${\mathbf{T}}_1$-equivalence. This leads to a logarithmically plurisubharmonic functional

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

(11)

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

Step 2: Subharmonicity of maximal function generated by $\mathcal{J}$. The functional (11) generates for any fixed $\theta \in [-\pi ,\pi ]$ and $F^{\mu }\in {\mathrm{\Sigma }}_{Q,\theta }$ the maximal function

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

(12)

Its argument $t=F^{\mu }\left(0\right)$ runs over some domain $D_{\theta }\subset \mathbb{C}$. The supremum in (12) is taken 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).

One of the crucial steps in the proof of Theorem 1 is to establish that every $u_{\theta }\left(t\right)$ inherits from $\mathcal{J}$ subharmonicity in t. This is provided by the following lemma.

Lemma 4.

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 $F^{\mu }\in {\mathrm{\Sigma }}_{Q,\theta }$, 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 ${\mathsf{\Gamma }}_m$ 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 ${\mathsf{\Gamma }}_m$ 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 $F^{\mu }\in {\mathrm{\Sigma }}_{Q,\theta }$ 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 ${\mathsf{\Gamma }}_m$. 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 $\mathbf{B}\left({\mathsf{\Gamma }}_m\right)$ a countable dense subset

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

For any of its points ${\phi }_p$, the corresponding extremal Teichüller disk $\mathbb{D}\left({\phi }_p\right)$ joining this point with the origin of $\mathbf{B}\left({\mathsf{\Gamma }}_m\right)$ 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 ${\psi }_{p;m}\in A_1(\mathbb{D},{\mathsf{\Gamma }}_m),{\parallel {\psi }_{p;m}\parallel }_1=1$.

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 (12). It is defined and subharmonic on the domain $D_{\theta }$. The lemma follows. □

Step 3: Majorization on cover of ${\mathrm{\Sigma }}_{af}$ 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 (11) that is logarithmically subharmonic in some domain $D_0$ (which, in view of weak rotational homogeneity of J is 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 ${\mathrm{\Sigma }}_{af}$ 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 ${\mathrm{\Sigma }}_{af}$ in the underlying space $\mathbf{T}$. Denote this image by $G_{af}$. Its structure is described by the following

Lemma 2 implies that the image of the set ${\mathrm{\Sigma }}_{af}$ 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 ${\mathrm{\Sigma }}_{af}$ and $Fib\left({\mathrm{\Sigma }}_{af}\right)$, 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 ${\mathrm{\Sigma }}_{{\theta }_j}$ 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 ${\mathrm{\Sigma }}_Q$ (in terms of variable $z^{\prime }$).

Noting that $J\left(f\right)$ is a polynomial of the form (1), we set

$${\tilde{J}}_{{\theta }_m}\left(F_a\right)=\sum _{\left|\alpha \right|=n_0}^N{C}_{m_1,\dots ,m_s}{e}^{2i{m}_1\theta }{a}_{m_1}^{{\alpha }_{m_1}}\dots e^{2i{m}_s\theta }{a}_{m_s}^{{\alpha }_{m_s}}.$$

This yields

$$\underset{m}{max}|{\tilde{J}}_{{\theta }_m}\left(F_a\right)|=\underset{{\sum }_{{\theta }_m}}{max}\left|\sum _{\left|\alpha \right|=n_0}^N{C}_{m_1,\dots ,m_s}{a}_{m_1}^{{\alpha }_{m_1}}\dots a_{m_s}^{{\alpha }_{m_s}}\right|=\underset{\mathrm{\Sigma }}{max}|\tilde{J}\left(F_f\right)|=\underset{S}{max}\left|J\left(f\right)\right|.$$

(13)

Similar relations are valid for the corresponding collections ${\mathrm{\Sigma }}_{af,{\theta }_j}$ 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 pick the sequence of increasing products of the quotient spaces

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

(14)

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 (13), 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

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

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

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)

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

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 (13),

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

(15)

The image of the set ${\mathrm{\Sigma }}_{af}$ 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 ${\mathrm{\Sigma }}_{af}$ 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 (14) 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 symmetry of the set ${\mathrm{\Sigma }}_{af}$ and of its image in ${\mathcal{T}}_m$ and from Lemma 2 (ensuring the existence of the needed values of $F^{\mu }\left(0\right)$).

It follows that every function $u_m\left(t\right)$ is a circularly symmetric function on its disk ${\mathbb{D}}_{a_m}$, and so is the 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 ${\mathrm{\Sigma }}_{af}$ 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 $F^{\mu }\in {\mathrm{\Sigma }}_{af}$ 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 ${\kappa }_{\theta }\left(z\right)$.

The same result is valid for the normalized functional $J^0\left(f\right)=J\left(f\right)/{max}_S\left|J\left(f\right)\right|$. Note also that the homotopy (and simultaneously extremal) disk of the function ${\kappa }_{\theta }$ is located entirely in the set ${\mathrm{\Sigma }}_{af}$

Step 4: Extremality of ${\kappa }_{\theta }$ on S. It remains to establish that $F_{\theta }$ is extremal on the whole class $\mathrm{\Sigma }$, which is equivalent to extremality of ${\kappa }_{\theta }$ on S.

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 via (5) 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

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

This metric majorates all conformal metrics determined by holomorphic maps $h:\mathbb{D}\to \mathbf{T}$ (via pull backing of ${\lambda }_{\mathbb{D}}$), in particular by $h\left(t\right)=f_t\left(z\right),f\in S$, on their holomorphic disks $h\left(\mathbb{D}\right)\subset \mathbf{T}$ and on $\mathbb{D}$; so

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

(16)

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

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

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

The relations (6) and (16), together with holomorphy with respect to $S_f$, imply that the integral distance on $\mathbb{D}$ generated by ${\lambda }_J$ satisfies

$$J\left(F_t\right)=J\left(F_{b_0,b_1;t}\right)+O\left(t^{p-1}\right)\mathrm{as}t\to 0,$$

and therefore, in view of extremality of ${\kappa }_{\theta }$ on ${\mathrm{\Sigma }}_{af}$,

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

(17)

with equality (even on one pair $(t,z)$) only if $f\left(z\right)={\kappa }_{\theta }\left(z\right)$. This provides the desired inequality

$$\underset{S}{max}\left|J\left(f\right)\right|\le |J\left({\kappa }_{\theta }\right)|,$$

with the same case of equality as in (17), completing the proof of Theorem 1.