Distortion Theory of Biunivalent Holomorphic Functions

Abstract

Biunivalent holomorphic functions form an interesting class in geometric function theory and are associated with special functions and solutions of complex differential equations. This paper provides a complete distortion theory for such functions, in particular, the sharp coefficient estimates. Another interesting feature is that (in contrast to general collections of univalent functions) one obtains in the same fashion the sharp bounds for coefficients of biunivalent functions with k-quasiconformal extension for any $k<1$.

1. Preamble

Biunivalent functions form an interesting subclass of holomorphic functions in the unit disk $\mathbb{D}=\left\{\right|z|<1\}$, which consists of univalent functions $w=f\left(z\right)=z+a_2{z}^2+\dots$ in $\mathbb{D}$ whose inverse functions $z=f^{-1}\left(w\right)$ also are univalent in this disk. Biunivalence causes rather strong rigidity.

There is an intrinsic connection of biunivalent functions with special functions, solutions of complex differential equations, with the so-called q-calculus, etc., where biunivalence naturally arises. From these points of view, biunivalent functions have been and remain intensively investigated by many authors, who have considered and defined new special subclasses of such functions depending on different parameters; see, e.g., [1,2,3,4,5,6,7,8,9] and the references cited there. These investigations resulted mainly in the estimates of the initial Taylor coefficients $a_2$ and $a_3$ and some of their combinations.

The aim of this paper is to show that the remarkable features of biunivalent functions allow one to create a complete distortion theory for this class, giving, in particular, the sharp coefficient estimates.

Estimating the holomorphic functionals depending on coefficients of univalent and more general holomorphic functions has the classical origins but still remains an important and intensively investigated field of geometric function theory with deep applications in Mathematics and Physics.

2. General Distortion Theorem for Biunivalent Functions

2.1. Preliminary Remarks

Consider in the class $\mathcal{B}$ the general rotationally homogeneous polynomial coefficient functionals

$$J\left(f\right)=J(a_{m_1},\dots ,a_{m_s}),2<a_{m_1}<\cdots <a_{m_s}<\infty ,$$

(1)

depending on a distinguished finite set of coefficients $a_{m_j}\left(f\right)$ such that $|J\left(f_{\alpha ,b}\right)|=|J\left(f\right)|$ for all pre- and post-rotations $w\mapsto e^{i\alpha }$ about the origin; here

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

(2)

Note that univalent functions are naturally connected with such rotations, because Schwarz’s lemma implies that for any $f\in S$ the image domain $f\left(\mathbb{D}\right)$ must have, with the unit circle ${\mathbb{S}}^1=\left\{\right|z|=1\}$, a common boundary point $z_0=e^{i{\beta }_0}$. This value is attained at a unique point $z_1=e^{i{\alpha }_0}$; so, using the rotations (2), one obtains a representative of f in the class S (in the general case, the correspondence between the boundary points $z_0$ and $z_1$ must be understood in the sense of the Carathéodory prime ends).

In the case of biunivalent functions, the curve $f\left({\mathbb{S}}^1\right)$ touches the circle ${\mathbb{S}}^1$ outside. Put

$$|a_2\left(\mathcal{B}\right)|=sup\{|a_2\left(f\right)|:f\in \mathcal{B}\},$$

(3)

and consider the set of rotations

$${\mathcal{R}}_{\mathcal{B}}=\{f_{0,\tau ,\theta }\left(z\right)=e^{-i\theta }{f}_0\left(e^{i\tau }z\right)\},$$

where $f_0$ is one of the maximizing functions for $a_2$ on $\mathcal{B}$ (their existence follows from compactness of $\mathcal{B}$ in topology of locally uniform convergence in $\mathbb{D}$).

E. Netanyahu established that $|a_2\left(\mathcal{B}\right)|=4/3$ [6]. We shall call any maximizing function for (3) the Netanyahu function.

2.2. Main Theorem

Theorem 1.

Any rotationally invariant polynomial functional (1), whose zero set ${\mathcal{Z}}_J=\{f\in \mathcal{B}:J\left(f\right)=0\}$ is separated from the rotation set ${\mathcal{R}}_{\mathcal{B}}$, is maximized only by the Netanyahu functions $f_{0,\tau ,\theta }$.

So, similar to the entire class S, any extremal function $f_0$ of any rotationally homogeneous coefficient functional J on $\mathcal{B}$ must be simultaneously maximal for the second coefficient $a_2$ on this class, unless $J\left(f_0\right)=0$.

The separation of sets indicated in Theorem 1 means that these sets are disjoint.

The general distortion results of such a type (in particular, strengthening the famous de Branges’ theorem [10] proving the Bieberbach conjecture) have been established recently by the author in [11,12,13] for rather general collections $\mathcal{X}$ of univalent functions on the disk satisfying the following two conditions:

$\left(a\right)$ openness, which means that the corresponding collection of the Schwarzians $S_w$ of $w\in \mathcal{X}$ determines in the space $\mathbf{T}$ a complex Banach submanifold (of finite or infinite dimension);

$\left(b\right)$ variational stability, which means that for any quasiconformal deformation h of a function $w\in \mathcal{X}$ with Beltrami coefficient ${\mu }_h$ (and sufficiently small dilatation ${\parallel \mu \parallel }_{\infty }$) supported in the complementary domain of $w\left(\mathbb{D}\right)$, the composition $h\circ w|\mathbb{D}$ also belongs to $\mathcal{X}$ (in fact, only a special type of such quasiconformal deformations has been applied).

Our first aim is to establish that the collection $\mathcal{B}$ of biunivalent functions obeys both these conditions; this yields that the image of $\mathcal{B}$ in $\mathbf{T}$ is a subdomain in this space, and one can apply to $\mathcal{B}$ the technique developed in [11,12,14].

This approach involves lifting the given functional $J\left(f\right)$ from S onto the universal Teichmüller space $\mathbf{T}$ and to its covering Bers’ fiber space $Fib\mathbf{T}$.

3. Rotational Extension of Univalent and Biunivalent Functions

3.1. General Univalent Functions

The general canonical classes of univalent functions are the classes S of functions

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

on the unit disk $\mathbb{D}=\left\{\right|z|<1\}$ and the class $\Sigma$ formed by univalent $\widehat{\mathbb{C}}$-holomorphic functions with the hydrodynamical normalization

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

on the complementary disk ${\mathbb{D}}^{*}=\{z\in \widehat{\mathbb{C}}=\mathbb{C}\cup \{\infty \}:\left|z\right|>1\}$. The unit circle $\left\{\right|z|=1\}$ will be denoted by ${\mathbb{S}}^1$.

The inversions $F_f\left(z\right)=1/f(1/z)$ of $f\in S$ are nonvanishing (zero-free) on ${\mathbb{D}}^{*}$. Let $S_Q$ and ${\Sigma }_Q$ denote their (dense) subclasses formed by functions admitting quasiconformal extension to the whole Riemann sphere $\widehat{\mathbb{C}}$.

The Beltrami coefficients ${\mu }_f\left(z\right)={\partial }_{\overline{z}}f/{\partial }_{z}f$ of these extensions run over the unit ball

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

and over the corresponding ball $Belt{\left(\mathbb{D}\right)}_1$, accordingly.

We consider more general classes $S_{Q,\theta }$ consisting of univalent functions

$$f\left(z\right)=a_{1}z+a_2{z}^2+\dots$$

(4)

on $\mathbb{D}$ having quasiconformal extensions across the unit circle ${\mathbb{S}}^1$ (hence to the whole sphere $\widehat{\mathbb{C}}$) and put

$$S^0=\bigcup _{\theta \in [-\pi ,\pi ]}{S}_{Q,\theta }$$

with topology of locally uniform convergence on the disk $\mathbb{D}$. The set $S^0$ is compact in this topology; the coefficients of its functions have the same moduli as the functions from S (being obtained from one another by the additional rotations of type (2)).

All functions $f_{\alpha ,\beta }\left(z\right)$ obtained from a given function $f\in S_{Q,\theta }$ by rotations (2) 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 $w=\gamma \left(z\right)$, 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 $\widehat{\mathbb{C}}$.

The Schwarzians $S_f$ belong to the complex Banach space $\mathbf{B}\left(\mathbb{D}\right)$ of hyperbolically bounded holomorphic functions $\phi$ in $\mathbb{D}$ with norm

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

and run over a bounded domain in the space $\mathbf{B}\left(\mathbb{D}\right)$. This domain models the universal Teichmüller space $\mathbf{T}=Teich\left(\mathbb{D}\right)$.

3.2. Rotationally Invariant Collection of Biunivalent Functions

In a similar fashion, one can rotationally extend the subclass $\mathcal{B}$, obtaining the classes ${\mathcal{B}}_{Q,\theta }$ of functions with expansions (4). The union

$${\mathcal{B}}^0=\bigcup _{\theta \in [-\pi ,\pi ]}{\mathcal{B}}_{Q,\theta }$$

consists of functions, which are univalent on the unit disk together with their inverse functions $f^{-1}$. Thus we shall call all functions from $\mathcal{B}$ biunivalent.

4. Digression to Teichmüller Spaces

We briefly recall some needed results from Teichmüller space theory on spaces involved in order to prove our main theorems. The details can be found, for example, in [15,16,17,18].

This theory is intrinsically connected with univalent functions with quasiconformal extension, and it is technically more convenient to deal with functions from ${\Sigma }_Q$. Note also that quasiconformal maps require three normalization conditions to have uniqueness, compactness, holomorphic dependence on parameters, etc.

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 another normalization of quasiconformally extendable functions.

Lemma 1.

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

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

(5)

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

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

(6)

and with more general normalization,

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

Observe also that the map $w\left(z\right)$ satisfying (4) is the elliptic fractional linear transformation

$$w={\displaystyle \frac{e^{i\theta }z}{(1-e^{i\theta })z_0^{-1}z+1}}$$

with fixed points 0 and $z_0$.

It follows from Lemma 1 that for any fixed ${\theta }_0\in [-\pi ,\pi ]$ there is a point $z_0=e^{i{\alpha }_0}\in {\mathbb{S}}^1$ such that for all $\theta$ with $|e^{i\theta }-z_0|<1$ any two Beltrami coefficients ${\mu }_1,{\mu }_2\in Belt{\left({\mathbb{D}}^{*}\right)}_1$ generate quasiconformal maps $w^{{\mu }_1}$ and $w^{{\mu }_2}$ normalized by (5) (hence, having the same fixed point $z_0$), unless these maps are conjugated by a rotation, or equivalently, ${\mu }_2\left(z\right)={\mu }_1\left(e^{i\alpha }z\right)e^{-2i\alpha }$ with some $\alpha \in [-\pi ,\pi ]$.

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 $\Gamma$, 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

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

over the universal Teichmüller space $\mathbf{T}$ with holomorphic projection $\pi (\psi ,z)=\psi$ (see [19,20]).

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

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

(7)

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 $\widehat{\mathbb{C}}$) 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 $z_j\in \mathbb{C}\setminus \{0,1\}$ (the details see, e.g., in [13]).

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 ${\mathbb{C}}^{n-3}$.

5. Proof of Theorem 1

We establish the assertion of this theorem in three stages.

• Step 1: Openness. First we prove the following underlying theorem, which provides openness of the image of $\mathcal{B}$ in the space $\mathbf{T}$. Denote this image by $G_{\mathcal{B}}$.

Theorem 2.

Any Schwarzian $S_{f_0}\in \mathbf{T}$ representing a biunivalent function $f_0\in \mathcal{B}$ has a neighborhood $U_0$ filled entirely by the Schwarzians of biunivalent functions. Hence, the collection of all Schwarzians corresponding to biunivalent functions is a subdomain in $\mathbf{T}$.

This theorem ensures, in particular, that biunivalence is preserved under quasiconformal deformations of $f\in \mathcal{B}$ with dilatations supported on the complementary domains $f\left({\mathbb{D}}^{*}\right)$ changing appropriately the second coefficient $a_2$.

• The proof of Theorem 2 is based on the following results presented via three lemmas. These results reveal a deep connection between biunivalence and geometry of Teichmüller balls.

Lemma 2

([14]). 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

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

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 the disk entirely: $D_{1/\left(2\right|a_{2,\theta }^0\left|\right)}=\left\{\right|w|<1/\left(2\right|a_{2,\theta }^0\left|\right)\}$.

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 $\left\{\right|W+a_{2,\theta }^0|\le |a_{2,\theta }^0\left|\right\}$.

The proof of this lemma follows the lines of the classical Koebe one-quoter theorem.

Lemma 3

([21]). Any holomorphic function $f\left(z\right)$ on the disk $\mathbb{D}$, whose Schwarzian $S_f$ lies in the ball $B_{\mathbf{T}}(\mathbf{0},\kappa )$ of radius $\kappa \le 1/4$ admits κ-quasiconformal extension onto $\widehat{\mathbb{C}}$ and is of the form

$$f\left(z\right)=\gamma \circ f_0\left(z\right),$$

where $f_0$ is biunivalent on $\mathbb{D}$ and γ is a Moebius transformation of $\widehat{\mathbb{C}}$. The upper-bound $1/4$ is sharp (cannot be increased).

Let $L\subset \mathbb{C}$ be an oriented bounded quasicircle separating the points 0 and ∞. Denote its interior and exterior domains by D and $D^{*}$ (so $0\in D,\infty \in D^{*}$). Then, if ${\delta }_D\left(z\right)$ denotes the Euclidean distance of z from the boundary of D and ${\lambda }_D\left(z\right)\left|dz\right|$ is its hyperbolic metric of Gaussian curvature $-4$, we have

$${\displaystyle \frac{1}{4}}\le {\lambda }_D\left(z\right){\delta }_D\left(z\right)\le 1.$$

Recall that

$${\lambda }_D\left(z\right)=h_{*}{\lambda }_{\mathbb{D}}\left(z\right):={\displaystyle \frac{|h^{\prime }\left(z\right)|}{1-{\left|h\left(z\right)\right|}^2}}\left|dz\right|,$$

where h is a conformal map of D onto the unit disk (see, e.g., [22,23]).

The Schwarzian derivatives of univalent functions in D belong to the complex Banach space $\mathbf{B}\left(D\right)$ of hyperbolically bounded holomorphic functions in D with norm

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

and run over a bounded domain in the space $\mathbf{B}\left(D\right)$. This domain models the same universal Teichmüller space $\mathbf{T}$ with base point D. The corresponding space $\mathbf{B}\left(D^{*}\right)$ for the complement domain $D^{*}$ is defined similarly. Note that each $\phi \in \mathbf{B}\left(D^{*}\right)$ satisfies $\phi \left(z\right)=O\left(\right|z{|}^{-4})$ as $z\to \infty$.

Lemma 4

([19]). There exists for some $\epsilon >0$ an anti-holomorphic homeomorphism τ (with $\tau \left(\mathbf{0}\right)=\mathbf{0}$) of the ball $V_{\epsilon }=\{\phi \in \mathbf{B}\left(D^{*}\right):\parallel \phi \parallel <\epsilon \}$ into $\mathbf{B}\left(D\right)$ such that every φ in $V_{\epsilon }$ is the Schwarzian derivative of some univalent function f, which is the restriction to $D^{*}$ of a quasiconformal automorphism $\widehat{f}$ of Riemann sphere $\widehat{\mathbb{C}}$. This $\widehat{f}$ can be chosen in such a way that its Beltrami coefficient is harmonic on D, i.e., of the form

$${\mu }_{\widehat{f}}\left(z\right)={\lambda }_D^{-2}\left(z\right)\overline{\psi \left(z\right)},\psi =\tau \left(\phi \right).$$

Since by assumption the function $f_0$ is biunivalent, the domain $f_0\left(\mathbb{D}\right)$ must contain the unit disk $\mathbb{D}$, and in view of the holomorphic contractibility of the hyperbolic metric, the $\mathbf{B}$-norm of restriction of the inverse function $f^{-1}$ to this disk satisfies

$$\parallel f^{-1}{\parallel }_{\mathbf{B}\left(\mathbb{D}\right)}\le {\parallel f^{-1}\parallel }_{\mathbf{B}\left(f\right(\mathbb{D}\left)\right)}<\epsilon .$$

(8)

We also pass to restrictions of all elements $\phi \in \mathbf{B}\left(f\right(\mathbb{D}\left)\right)$ to $\mathbb{D}$.

By the Ahlfors–Weill theorem [24], every function $\phi \in \mathbf{B}\left(\mathbb{D}\right)$ with ${\parallel \phi \parallel }_{\mathbf{B}}<2$ is the Schwarzian derivative of a univalent function $f\left(z\right)$ on $\mathbb{D}$, and the function f has quasiconformal extension onto the disk ${\mathbb{D}}^{*}$ with Beltrami coefficient

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

Hence, the extremal extension ${\widehat{f}}^{-1}$ of $f^{-1}$ onto ${\mathbb{D}}^{*}$ has in ${\mathbb{D}}^{*}$ the Beltrami coefficient ${\mu }_{wh{f}^{-1}}$ with

$$\parallel {\mu }_{wh{f}^{-1}}{{\parallel }_{\infty }\le {\mu }_{{\phi }_0}\parallel }_{\infty }<\epsilon .$$

Now choose among the functions with the same Schwarzian derivative $\phi =S_F$ the function normalized additionally by $F(\infty )=\infty$. For such functions with k-quasiconformal extensions, we have the well-known sharp bound

$$|a_2|\le 2k,$$

with equality only for the function

$$f_k\left(z\right)=z/{(1-tkz)}^2,\left|z\right|\le 1,\left|t\right|=1,$$

whose extremal Beltrami coefficient among quasiconformal extensions across the unit circle to $\widehat{\mathbb{C}}$ has the form ${\mu }_0\left(z\right)=k{\left|z\right|}^3/z^3$.

Together with Lemmas 2 and 3, this provides the assertion of Theorem 2 on the existence of (a small) neighborhood $U_0$ filled by the images of biunivalent functions (openness of the set $G_{\mathcal{B}}$) follows from these lemmas immediately.

To establish that set $G_{\mathcal{B}}$ is pathwise connected, we pass the homotopy functions

$$f_r\left(z\right)={\displaystyle \frac{1}{r}}f\left(rz\right)\mathrm{with}r\in [0,1](f\in S).$$

Their Schwarzians $S_{f_r}\left(z\right)={\displaystyle \frac{1}{r^2}}{S}_f\left(rz\right)$ depend holomorphically on $(z,r)\in \mathbb{D}\times \mathbb{D}$ in C-norm and by the well-known properties of holomorphic functions with sup-norm also in $\mathbf{B}$-norm.

This yields that each Schwarzian $S_f$ representing a biunivalent function can be connected with the origin of $\mathbf{T}$ by a continuous path consisting only of such functions, which completes the proof of Theorem 2.

• We proceed to the proof of Theorem 1 whose second step consists of lifting the functional $J\left(f\right)$ to $\mathbf{T}$ and $Fib\left(\mathbf{T}\right)$.

For a technical convenience, we now pass to the inverted functions $F_f\left(z\right)=1/f(1/z)$ for $f\in S$, which form the corresponding classes ${\Sigma }_{z_0,\theta }$ of nonvanishing (zero-free) univalent functions on the disk ${\mathbb{D}}^{*}$ with expansions

$$F\left(z\right)=e^{-i\theta }z+b_0+b_1{z}^{-1}+b_2{z}^{-2}+\dots ,F\left(z_0\right)=z_0,$$

and set

$${\Sigma }^0=\bigcup _{z_0,\theta }{\Sigma }_{z_0,\theta }.$$

The coefficients $a_n$ of $f\left(z\right)$ and the corresponding coefficients $b_j$ of $F_f\left(z\right)$ are related by

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

where ${ϵ}_{n,j}$ are the entire powers of $e^{i\theta }$. This successively implies the representations of $a_n$ by $b_j$ via

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

(9)

This transforms the initial functional (1) into a coefficient functional $\tilde{J}\left(F^{\mu }\right)$ on ${\Sigma }^0$ depending on the corresponding coefficients $b_j$. The relations (9) imply that $\tilde{J}\left(F^{\mu }\right)$ depends holomorphically on the Beltrami coefficients ${\mu }_F\in Belt{\left(\mathbb{D}\right)}_1$ and on the corresponding Schwarzians $S_F$.

Accordingly, we model the universal Teichmüller space $\mathbf{T}$ by a domain in the space $\mathbf{B}=\mathbf{B}\left({\mathbb{D}}^{*}\right)$ formed by the Schwarzians $S_F^{\mu }$ with $\mu$ running over the Beltrami ball $Belt{\left(\mathbb{D}\right)}_1$. Thereby, both functionals $\tilde{J}\left(F\right)$ and $J\left(f\right)$ are lifted holomorphically onto the space $\mathbf{T}$.

The next step is to lift these functionals onto the covering space ${\mathbf{T}}_1$. To get this, we again pass to the functional $\widehat{J}\left(\mu \right)=\tilde{J}\left(F^{\mu }\right)$ lifting it onto the ball $Belt{\left(\mathbb{D}\right)}_1$ and apply now the ${\mathbf{T}}_1$-equivalence, i.e., the quotient map

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

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

Now. applying the Bers isomorphism theorem, we can regard the points of the space ${\mathbf{T}}_1\cong Fib\left(\mathbf{T}\right)$ 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 (hence, also $\mathbf{T}$-equivalence). Note that since the coefficients $b_0,b_1,\dots$ of $F^{\mu }\in {\Sigma }_{\theta }$ are uniquely determined by its Schwarzian $S_{F^{\mu }}$, the values of $\mathcal{J}$ in the points $X_1,X_2\in {\mathbf{T}}_1$ with ${\iota }_1\left(X_1\right)={\iota }_1\left(X_2\right)$ are equal, where $\iota$ is the holomorphic map ${\iota }_1:{\mathbf{T}}_1\to \mathbf{T}$ generated by the inclusion ${\mathbb{D}}_{*}\hookrightarrow \mathbb{D}$ forgetting the puncture.

As a result, we get on ${\mathbf{T}}_1=Fib\left(\mathbf{T}\right)$ the holomorphic functional

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

and must investigate the restriction of the plurisubharmonic functional $\left|\mathcal{J}\right(S_{F^{\mu }},t\left)\right|$ to the image of domain $G_{\mathcal{B}}$ in $Fib\left(\mathbf{T}\right)$.

By Lemma 2, the boundary of domain $W^{\mu }\left({\mathbb{D}}^{*}\right)$ under any function $W^{\mu }\left(z\right)\in {\Sigma }^0$ is located in the disk $\left\{\right|W-b_0|\le |a_2\left(\mathcal{B}\right)\left|\right\}$, and the variable t in the representation (10) runs over some subdomain $D_1$ in the disk ${\mathbb{D}}_4=\left\{\right|t|<4\}$ containing the origin (this subdomain depends on $z_1$). Since the functional J is rotationally invariant, this subdomain $D_1$ is a disk ${\mathbb{D}}_{{\alpha }_1}$ of some radius ${\alpha }_1\le 2\left|a_2\left(\mathcal{B}\right)\right|$.

We define on this domain the function

$${\tilde{u}}_1\left(t\right)=\underset{S_{W^{\mu }}}{sup}\mathcal{J}(S_{W^{\mu }},t)$$

(10)

taking the supremum over all $S_{W^{\mu }}\in \mathbf{T}$ admissible for a given $t=W^{\mu }\left(0\right)\in D_{{\alpha }_n}$, that means over the pairs $(S_{W^{\mu }},t)\in Fib\left(\mathbf{T}\right)$ with a fixed t and pass to the upper semicontinuous regularization

$$u_1\left(t\right)=\underset{t^{\prime }\to t}{lim\; sup}{\tilde{u}}_1\left(t^{\prime }\right).$$

Now the crucial step in the proof of Theorem 1 is to establish that the function (9) inherits subharmonicity. In fact, we have much more.

Namely, selecting on the unit circle a dense subset

$$\mathbf{e}=\{z_1,z_2,\dots ,z_n,\dots \},z_1=e^{i{\theta }_1},$$

and repeating successively for the above construction with fixed points $z_1,z_2,\dots$, one obtains, similar to (10), the corresponding functions $u_1\left(t\right),u_2\left(t\right),\dots$. Let $u\left(t\right)$ be their upper envelope ${sup}_n{u}_n\left(t\right)$ followed by its upper semicontinuous regularization.

Lemma 5.

The function $u\left(t\right)$ is logarithmically subharmonic in the disk ${\mathbb{D}}_{\alpha }$ with $\alpha =2|a_2\left(\mathcal{B}\right)|$.

In view of assumptions on the domain $G_{\mathcal{B}}$, this lemma is a consequence of the general basic Lemma 1 from [13], which provides subharmonicity of the maximal function $u\left(t\right)$ as the supremum of $\mathcal{J}(S_{F^{\mu }},t))$ over all $S_{F^{\mu }}$ running on the whole space $\mathbf{T}$.

Theorem 1 relates to the restriction of this function onto the image of $G_{\mathcal{B}}$ in $Fib\left(\mathbf{T}\right)$. By Theorem 2, this image also is a Banach domain. Hence, the basic properties of plurisubharmonic functions (in Euclidean and Banach domains) imply that the indicated restriction of the functional $\mathcal{J}(S_{F^{\mu }},t)$ onto this domain also is plurisubharmonic, which implies the subharmonicity of the constructed function $u\left(t\right)$ on some disk.

• Step 3: Finishing the proof. It remains to establish the range domain of $F^{\mu }\left(0\right)$ for $S_{F^{\mu }}$ running over $G_{\mathcal{B}}$ and describe the boundary points of this domain. This step straightforwardly follows the lines of [13] varying the functional $\mathcal{J}(S_{F^{\mu }},t)$ quasiconformally on domains $F^{\mu }\left(\mathbb{D}\right)$. Theorem 2 implies that these variations (changing appropriately the coefficient $a_2$) are admissible (preserve biunivalence).

By the previous step, the desired range domain is rotationally symmetric, hence a disk ${\mathbb{D}}_{\alpha }=\left\{\right|t|<\alpha \}$ of some radius

$$\alpha \le 2|a_2\left(\mathcal{B}\right)|.$$

Lemma 2 implies that this is a disk of maximal radius, which equals to $2|a_2\left(\mathcal{B}\right)|$.

Hence the maximum of $\left|J\right|$ is attained on the boundary circle $\left\{\right|t|=2|a_2\left(\mathcal{B}\right)|$, and the assertion of Theorem 1 follows.

6. Biunivalent Functions with $\mathit{k}$-Quasiconformal Extension

6.1. New Features of Functions with k-Quasiconformal Extension

Another interesting fact related to biunivalent functions is that (in contrast to the general collections of normalized univalent functions) the subclasses ${\mathcal{B}}_k$ of $\mathcal{B}$ consisting of functions with k-quasiconformal extensions with a fixed $k<1$ also obey both properties $\left(a\right),\left(b\right)$ indicated after Theorem 1. This allows us to estimate sharply the coefficients of any class ${\mathcal{B}}_k$. Namely, letting

$$|a_2\left({\mathcal{B}}_k\right)|=\underset{f\in {\mathcal{B}}_k}{sup}\left|a_2\left(f\right)\right|,$$

we have the following.

Theorem 3.

Let $w=f_k\left(z\right)$ be a maximizing function for $|a_2\left({\mathcal{B}}_k\right)|$. Then any rotationally invariant polynomial functional (1) on ${\mathcal{B}}_k$, whose zero set ${\mathcal{Z}}_J$ is separated from the rotation set $\{f_{k,\tau ,\theta }=e^{-i\theta }{f}_k\left(e^{i\tau }z\right)\}$ is maximized only by these rotations.

Proof. 

One only has to establish that the classes ${\mathcal{B}}_k$ admit the variational stability in the sense of the condition $\left(b\right)$, because the openness was already established for the entire class $\mathcal{B}$. The point now is that one can only use the quasiconformal variations of $f\in {\mathcal{B}}_k$ preserving the maximal dilatation k.

Such variations have been constructed for the general quasiconformal maps and for univalent functions with quasiconformal extensions in the book [25] in terms of the inverse maps $z=f^{-1}\left(w\right)$; see there Chapters 2–4. In addition, the above Theorem 2 implies that these $\epsilon$-quasiconformal variations are admissible for sufficiently small $\epsilon$ in the sense that these variations preserve biunivalence.

Having biunivalence (which provides that both functions f and $f^{-1}$ simultaneously belong to ${\mathcal{B}}_k$), one can apply the technique from [25] directly to the original functions $f\left(z\right)$ (cf. [14,26,27]) and obtain the assertion of Theorem 3 in the same fashion as Theorem 1. □

6.2. Question

To what extent the above results are prolonged to holomorphic functionals on classes of biunivalent functions admitting only the weak homogeneity (2) with $\beta =\alpha$?

One of the important general properties of extremal univalent functions $f_0\in S$ with k-quasiconformal extensions maximizing the functionals $J\left(f\right)$ in this class is also that the boundary of the extremal domain $f_0\left(\mathbb{D}\right)$ is determined by an extremal Beltrami coefficient of this function. Such a fact is not known for biunivalent functions.