Univalent functions with quasiconformal extensions: Becker's class and estimates of the third coefficient

TL;DR

This paper studies univalent functions with quasiconformal extensions, focusing on coefficient estimates, Becker's class, and the maximal dilatation for extensions, providing new bounds and optimality results.

Contribution

It introduces new bounds for the third coefficient of univalent functions with quasiconformal extensions and characterizes the maximal dilatation for Becker's class.

Findings

Established that the maximal $k_*$ for Becker extensions is at least 1/6.

Provided estimates for the third coefficient $a_3$ of univalent functions with q.c. extensions.

Showed that Becker's extension can be optimal in certain cases.

Abstract

We investigate univalent functions $f(z)=z+a_2z^2+a_3z^3+\ldots$ in the unit disk $\mathbb D$ extendible to $k$-q.c.(=quasiconformal) automorphisms of $\mathbb C$. In particular, we answer a question on estimation of $|a_3|$ raised by K\"uhnau and Niske [Math. Nachr. 78 (1977) 185-192]. This is one of the results we obtain studying univalent functions that admit q.c.-extensions via a construction, based on Loewner's parametric representation method, due to Becker [J. Reine Angew. Math. 255 (1972) 23-43]. Another problem we consider is to find the maximal $k_*\in(0,1]$ such that every univalent function $f$ in $\mathbb D$ having a $k$-q.c. extension to $\mathbb C$ with $k\leqslant k_*$ admits also a Becker q.c.-extension, possibly with a larger upper bound for the dilatation. We prove that $k_*>1/6$. Moreover, we show that in some cases, Becker's extension turns out to be the optimal one. Namely, given any $k\in(0,1)$, to each finite Blaschke product there corresponds a univalent function $f$ in $\mathbb D$ that admits a Becker $k$-q.c. extension but no $k'$-q.c. extensions to $\mathbb C$ with $k'<k$.

Figures

Equations

$$\mathbb{D}\ni z\mapsto f(z)=z+\sum _{n=2}^{+\infty }{a}_n{z}^n.$$

$$\frac{\mathrm{d}w}{\mathrm{d}t}=-wp(w,t),t⩾0,w(z,0)=z\in \mathbb{D},$$

$$f(z):=\underset{t\to +\infty }{\lim }\frac{w(z,t)}{w^{\prime }(0,t)},z\in \mathbb{D},$$

$${\int }_0^{+\infty }\mathrm{Re}p(0,t)\mathrm{d}t=+\infty ,$$

$$p(\mathbb{D},t)\subset U(k):=\left\{w\in \mathbb{C}:\left|\frac{w-1}{w+1}\right|⩽k\right\}\text{for a.e. t⩾0}.$$

$$F(\rho e^{i\theta}):=\left\{\begin{array}[]{ll}f_{0}(\rho e^{i\theta}),&\text{if~{}$0\leqslant\rho<1$},\\[4.30554pt] f_{\log\rho}(e^{i\theta}),&\text{if~{}$\rho\geqslant 1$}.\end{array}\right.$$

$$|a_{3}|\leqslant k\big{(}1+e^{1-1/k}(1+k)\big{)}.$$

$$\max_{\mathcal{S}_{k}^{B}}|a_{3}|<\max_{\mathcal{S}_{k}}|a_{3}|\leqslant\varrho(k):=\min_{\alpha\in(0,1)}\Bigl{[}\big{(}1+2e^{-2\alpha/(1-\alpha)}\big{)}k+4\alpha k^{2}\Bigr{]}.$$

$${\mathcal{H}}_k\ni p\mapsto f:=\underset{t\to +\infty }{\lim }{e}^{t}w(z,t)\in \mathcal{S},$$

$$f(z,t):=e^{t}w(z,t)=z+a_2(t)z^2+a_3(t)z^3+\dots$$

$$L(z):=\frac{1+Kz}{K+z},K:=\frac{1+k}{1-k},$$

$$\frac{\mathrm{d}{a}_2}{\mathrm{d}t}$$

$$\frac{\mathrm{d}{a}_3}{\mathrm{d}t}$$

$$\displaystyle=-k\Big{(}e^{-2t}\big{(}c_{2}(t)-(1-k)\frac{c_{1}(t)^{2}}{2}\,\big{)}+2e^{-t}c_{1}(t)a_{2}(t)\Big{)},$$

$$|c_1|⩽2,|2c_2-c_1^2|⩽4-|c_1{|}^2.$$

$$H(a_{2},a_{3},\psi_{2},\psi_{3},t,c_{1},c_{2}):=-ke^{-t}c_{1}\psi_{2}-k\Big{(}e^{-2t}\big{(}c_{2}-(1-k)\frac{c_{1}^{2}}{2}\big{)}+2e^{-t}c_{1}a_{2}\Big{)}\psi_{3}$$

$$\frac{\mathrm{d}{\psi }_2}{\mathrm{d}t}$$

$$\frac{\mathrm{d}{\psi }_3}{\mathrm{d}t}$$

$${\psi }_2(+\infty )=0,{\psi }_3(+\infty )=1,$$

$$\max_{(c_{1},c_{2})}\mathop{\mathtt{Re}}H\big{(}a_{2}(t),a_{3}(t),\psi_{2}(t),\psi_{3}(t),t,c_{1},c_{2}\big{)}\\ =\mathop{\mathtt{Re}}H\big{(}a_{2}(t),a_{3}(t),\psi_{2}(t),\psi_{3}(t),t,c^{*}_{1}(t),c^{*}_{2}(t)\big{)},$$

$${\psi }_2(t)=a-2a_2(t),{\psi }_3(t)=1,$$

$$-\frac{e^{2t}}{k}\mathrm{Re}H$$

$$=\frac{1+k}{2}{\left(c_1^{\prime }+\frac{e^t{a}^{\prime }}{1+k}\right)}^2+\frac{1-k}{2}{\left(c_1^{\prime \prime }-\frac{e^t{a}^{\prime \prime }}{1-k}\right)}^2+C,$$

$$\mathrm{sgn}\mathrm{Re}{c}_1^{\ast }=-\mathrm{sgn}{a}^{\prime },\mathrm{sgn}\mathrm{Im}{c}_1^{\ast }=\mathrm{sgn}{a}^{\prime \prime },$$

$$a=-2k{\int }_0^{+\infty }{e}^{-t}{c}_1^{\ast }(t)\mathrm{d}t,$$

$$c_{1}^{*}(t)=\begin{cases}-e^{t}a/(1+k),&\text{if~{}}\big{|}e^{t}a/(1+k)\big{|}<2,\\ -2\mathop{\mathrm{sgn}}a,&\text{otherwise}.\end{cases}$$

$$p_0(z,t)$$

$$=1+2(\lambda {\mu }_1+(1-\lambda ){\mu }_2)z+2(\lambda {\mu }_1^2+(1-\lambda ){\mu }_2^2)z^2+\dots ,z\in \mathbb{D},$$

$$a=2k\left(\frac{a{t}_0}{1+k}+2e^{-t_0}\mathrm{sgn}a\right)=\frac{2k(1+t_0)}{1+k}a.$$

$$a_2(+\infty )=a/2=\pm \alpha (k),\text{where α(k):=(1+k)e−t0.}$$

$$a_3(+\infty )=a_2(+\infty {)}^2-\underset{0}{\overset{+\infty }{\int }}{e}^{-2t}{p}_2^{\ast }(t)\mathrm{d}t,$$

$$a_{3}(+\infty)=k\big{(}1+e^{1-1/k}(1+k)\big{)}>k=a_{3}|_{f=f_{1}}.$$

$$p(z,t)=\frac{1-k{z}^2+(1-k)e^{t-t_0}z}{1+k{z}^2+(1+k)e^{t-t_0}z}\text{ for t∈[0,t0]}\text{and}p(z,t)=\frac{1-kz}{1+kz}\text{ for t⩾t0}.$$

Full text

† Ikkei Hotta is supported by JSPS KAKENHI Grant Number 17K14205.

1. Introduction

Conformal mappings of $\mathbb{D}:=\{z:|z|<1\}$ admitting quasiconformal extensions is a classical topic in Geometric Function Theory closely related to Teichmüller Theory, see e.g. [32, 40]. Let $k\in(0,1)$. A function $f$ holomorphic in a domain $D\subset\mathbb{C}$ is said to be $k$-q.c. extendible to $\mathbb{C}$ (or to $\overline{\mathbb{C}}$) if there exists a $k$-quasiconformal automorphism ${F:\mathbb{C}\to\mathbb{C}}$ (respectively, ${F:\overline{\mathbb{C}}\to\overline{\mathbb{C}}}$) such that $F|_{\mathbb{D}}=f$. Note that $k$-q.c. extendibility to $\mathbb{C}$, which we will be mostly concerned with in this paper, is equivalent to $k$-q.c. extendibility to $\overline{\mathbb{C}}$ with the additional condition that $F(\infty)=\infty$.

Denote by $\mathcal{S}$ the class of all univalent (i.e. injective holomorphic) functions

[TABLE]

One of the main tools to study this class is the parametric representation, which goes back to Loewner [33], see e.g. [35, §6.1], see also [26, 34, 17]. Namely, the class $\mathcal{S}$ can be represented as an image of the convex cone formed by the so-called Herglotz functions, i.e. functions $p:\mathbb{D}\times[0,+\infty)\to\mathbb{C}$ such that $p(z,\cdot)$ is locally integrable for each $z\in\mathbb{D}$ and $p(\cdot,t)$ is holomorphic in $\mathbb{D}$ and satisfies $\mathop{\mathtt{Re}}p(\cdot,t)\geqslant 0$ for a.e. $t\geqslant 0$. It is known that for any Herglotz function $p$, the initial value problem for the Loewner – Kufarev ODE

[TABLE]

has a unique solution $w=w(z,t)$ and the locally uniform limit

[TABLE]

where $w^{\prime}$ denotes the derivative w.r.t. $z$, exists and belongs to $\mathcal{S}$. On the other hand, see e.g. [35, Theorem 6.1 on p. 159] or [17], every function $f\in\mathcal{S}$ can be represented by (1.2) with a suitable, and in general not unique, normalized Herglotz function, i.e. a Herglotz function $p$ with $\mathop{\mathtt{Re}}p(0,t)=1$ for a.e. $t\geqslant 0$.

A natural problem arises: given a subclass $\mathrlap{\hskip 1.25pt\widetilde{\hphantom{\mathcal{S}}\vphantom{\vbox to6.45831pt{\vss}}}}\mathcal{S}\vphantom{\vbox to9.90276pt{\vss}}\subset\mathcal{S}$, find a class of Herglotz functions that generates $\mathrlap{\hskip 1.25pt\widetilde{\hphantom{\mathcal{S}}\vphantom{\vbox to6.45831pt{\vss}}}}\mathcal{S}\vphantom{\vbox to9.90276pt{\vss}}$ via (1.2). The answer is known in some cases, e.g. for starlike functions, bounded univalent functions, and for univalent functions with real Taylor coefficients; see e.g. [38].

A partial answer is also known for the subclass $\mathcal{S}_{k}$, $k\in(0,1)$, formed by all $f\in\mathcal{S}$ admitting $k$-q.c. extension to $\mathbb{C}$. Namely, in 1972, Becker [3] found a condition on $p$ in the Loewner – Kufarev equation (1.1), see Sect. 2, such that the function $f$ given by (1.2) belongs to $\mathcal{S}_{k}$. The class $\mathcal{S}_{k}^{B}$ generated by Herglotz functions that satisfy Becker’s condition is a proper subset of $\mathcal{S}_{k}$. In this paper we study $\mathcal{S}_{k}^{B}$ and its relation with $\mathcal{S}_{k}$. In particular, in Sect. 3 we find the sharp estimate for $|a_{3}|$ in $S_{k}^{B}$, see Theorem 3.1. An immediate corollary is the answer to a question of Kühnau and Niske [28]: Theorem 3.1 implies that $\max_{S_{k}}|a_{3}|>k$ for any $k\in(0,1)$.

Numerous sharp estimates are known for the class $\mathcal{S}$, see e.g. [13], with many of them being motivated by the famous Bieberbach Conjecture concerning estimates for $|a_{n}|$, which was proved by de Branges [11] in 1984. Unfortunately, only a few of these results have been extended to classes $\mathcal{S}_{k}$, see e.g. [27, 30]. In particular, the sharp estimate for $|a_{n}|$ in $\mathcal{S}_{k}$ is known only for $n=2$. Remarkably, in most of the cases discussed previously, the extremal functions belong to $\mathcal{S}_{k}^{B}$. We prove a bit surprising fact that this does not hold for the sharp estimate of $|a_{3}|$, see Theorem 3.2.

2. Becker’s construction of quasiconformal extensions

Throughout the paper we make use of Loewner Theory, the classical version of which can be found in [35, Chapter 6]. Following Becker [4], [5, §5.1], we replace the usual normalization ${p(0,t)=1}$ by a weaker condition

[TABLE]

which still implies that $\bigcup\limits_{t\geqslant 0}f_{t}(\mathbb{D})=\mathbb{C}$. In 1972, he discovered the following remarkable fact.

Theorem A** ([3, 4]).**

Let $k\in[0,1)$ and let $(f_{t})$ be a radial Loewner chain whose Herglotz function $p$ satisfies

[TABLE]

Then for every $t\geqslant 0$, the function $f_{t}$ admits a $k$-q.c. extension to $\overline{\mathbb{C}}$ that fixes $\infty$. In particular, such an extension for $f_{0}$ is given by

[TABLE]

Remark 2.1*.*

According to [16, Theorem 2], a sort of converse statement holds. Namely, if $(f_{t})$ is a Loewner chain such that all $f_{t}$’s extend continuously to $\partial\mathbb{D}$ and the map $F$ defined by (2.3) is $k$-quasiconformal in $\mathbb{C}$, then the Herglotz function $p$ associated with $(f_{t})$ satisfies Becker’s condition (2.2).

In what follows, for $k\in(0,1)$, we will denote by $\mathcal{S}_{k}^{B}$ the class of all $f\in\mathcal{S}$ admitting Loewner’s representation with the Herglotz function $p$ normalized by $p(0,t)=1$ a.e. $t\geqslant 0$ and satisfying (2.2). A bit larger class of all $f\in\mathcal{S}$ generated by Herglotz functions subject to Becker’s condition (2.2), but not necessarily normalized, will be denoted by $\mathrlap{\hskip 1.25pt\widetilde{\hphantom{\mathcal{S}}\vphantom{\vbox to6.45831pt{\vss}}}}\mathcal{S}_{k}^{B}\vphantom{\vbox to9.90276pt{\vss}}$.

According to Theorem A, $\mathcal{S}^{B}_{k}\subset\mathrlap{\hskip 1.25pt\widetilde{\hphantom{\mathcal{S}}\vphantom{\vbox to6.45831pt{\vss}}}}\mathcal{S}_{k}^{B}\vphantom{\vbox to9.90276pt{\vss}}\subset\mathcal{S}_{k}$. It is known that $\mathrlap{\hskip 1.25pt\widetilde{\hphantom{\mathcal{S}}\vphantom{\vbox to6.45831pt{\vss}}}}\mathcal{S}_{k}^{B}\vphantom{\vbox to9.90276pt{\vss}}\neq\mathcal{S}_{k}$, see e.g. [16, §5]. However, it seems that the study of $\mathcal{S}_{k}^{B}$ and $\mathcal{S}_{k}$ is still of considerable interest. It is worth to mention that Becker’s condition (2.2) appears to be sufficient for q.c.-extendibility also in the framework of the general Loewner Theory introduced in [10, 9]; see [16], [21], and [15]. This discussion will be continued in Sect. 5.

3. Estimate of the third coefficient

Below we give a sharp estimate for $|a_{3}|$ in the class $\mathcal{S}_{k}^{B}$. As a corollary, we immediately obtain a negative answer to the question raised in 1977 by Kühnau and Niske [28]: does there exist $k_{0}>0$ such that for any $k\in(0,k_{0}]$ and any function $f(z)=z+a_{2}z+a_{3}z^{3}+\ldots$ belonging to $\mathcal{S}_{k}$, the inequality $|a_{3}|\leqslant k$ holds?

Theorem 3.1**.**

Let $k\in(0,1)$. Then for every function $f(z)=z+a_{2}z^{2}+a_{3}z^{3}+\ldots$ belonging to $\mathcal{S}_{k}^{B}$,

[TABLE]

This estimate is sharp and the equality holds only for rotations of the function $f_{+}\in\mathcal{S}_{k}^{B}$, which is uniquely defined by the Beltrami coefficient (3.15) of its q.c.-extension to $\mathbb{C}$.

The above theorem does not solve the extremal problem $|a_{3}|\to\max$ in the whole class $\mathcal{S}_{k}$. In fact, the following takes place.

Theorem 3.2**.**

For any $k\in(0,1)$,

[TABLE]

Remark 3.3*.*

The sharp estimate in Theorem 3.1 shows that the inequality ${|a_{n}|\leqslant 2k/(n-1)}$ written in the larger class $\mathcal{S}_{k}$ for $0<k\leqslant 1/(1+n^{2})$ and all $n=2,3,\ldots$ by Krushkal [25, Corollary on p. 350], in fact, fails for $n=3$. Note that the two estimates have tangency of infinite order at $k=0$, while the difference from the r.h.s. of (3.1) behaves asymptotically as $4k^{2}$. The three estimates are shown in Figure 1.

Proof of Theorem 3.1.

The class $\mathcal{S}_{k}^{B}$, ${k\in(0,1)}$, admits a Loewner-type parametric representation. Denote by $\mathcal{H}_{k}$ the class of all normalized Herglotz functions $p$ satisfying $p(\mathbb{D},t)\subset U(k)$ for a.e. $t\geqslant 0$, where $U(k)$ is the closed disk defined in Theorem A. As it follows from the very definition, $\mathcal{S}_{k}^{B}$ coincides with the image of the map

[TABLE]

where for each $z\in\mathbb{D}$ the function $[0,+\infty)\ni t\mapsto w(z,t)\in\mathbb{D}$ is defined as the unique solution to the initial value problem (1.1). Write $p(z,t)=1+p_{1}(t)z+p_{2}(t)z^{2}+\ldots$ for all $z\in\mathbb{D}$ and a.e. $t\geqslant 0$ and let

[TABLE]

There is one-to-one correspondence between the class of all normalized Herglotz functions and $\mathcal{H}_{k}$. Indeed, $p\in\mathcal{H}_{k}$ if and only if it can be written as $p(\cdot,t)=L\circ p_{0}(\cdot,t)$ for a.e. $t\geqslant 0$, where $p_{0}(z,t)=1+c_{1}(t)z+c_{2}(t)z^{2}+\ldots$ is an arbitrary normalized Herglotz function and

[TABLE]

is a conformal map of $\mathbb{H}:=\{z\colon\mathop{\mathtt{Re}}z>0\}$ onto $U(k)$ with $L(1)=1$.

As usual, from (1.1) we obtain the initial value problem for the coefficients $a_{2}$ and $a_{3}$,

[TABLE]

Since along with any $f\in\mathcal{S}_{k}^{B}$ the class $\mathcal{S}_{k}^{B}$ contains all rotations of $f$, i.e. the functions ${z\mapsto e^{i\theta}f(e^{-i\theta}z)}$, ${\theta\in\mathbb{R}}$, the problem to determine $\max|a_{3}|$ in $\mathcal{S}_{k}^{B}$ is equivalent to finding $~{}\max\,\mathop{\mathtt{Re}}a_{3}$. The latter problem can be reformulated as the optimal control problem for the above system and the objective functional $\mathop{\mathtt{Re}}a_{3}(+\infty)$, with a control function {t\mapsto\big{(}c_{1}(t),c_{2}(t)\big{)}\in\mathbb{C}^{2}} regarded as admissible if it is measurable and for a.e. $t\geqslant 0$ satisfies

[TABLE]

Conditions (3.4) describe the value region of $\mathcal{C}\ni q\mapsto(c_{1},c_{2})\in\mathbb{C}^{2}$ over the Carathéodory class $\mathcal{C}$ of all holomorphic functions $q(z)=1+c_{1}z+c_{2}z^{2}+\ldots$ in $\mathbb{D}$ with positive real part; see, e.g., [43, Chapter IV, §7].

To apply Pontryagin’s Maximum Principle, we define the (holomorphic) Hamiltonian

[TABLE]

and write the adjoint system

[TABLE]

The maximum of $\mathop{\mathtt{Re}}a_{3}(+\infty)$ is to be found among all the trajectories of (3.2), (3.3) satisfying the initial condition at $t=0$, while the right-hand endpoint of the trajectories is variable. Therefore, according to Pontryagin’s Maximum Principle, see [37, Chapter I, §7, Theorem $3^{*}$], if $c_{1}(t)=c_{1}^{*}(t)$, $c_{2}(t)=c_{2}^{*}(t)$ is an optimal control in our problem, then for the corresponding solution to the phase system (3.2), (3.3) supplemented with the adjoint equations (3.5), (3.6) and the transversality conditions

[TABLE]

it holds that

[TABLE]

where the maximum is taken over all $(c_{1},c_{2})\in\mathbb{C}^{2}$ subject to conditions (3.4).

System (3.5) – (3.7) can be integrated using integrals to (3.2), (3.3):

[TABLE]

where $a:=2a_{2}(+\infty)$.

To find the maximum of $\mathop{\mathtt{Re}}H$ as a function of $c_{1}$ and $c_{2}$, we first fix a $c_{1}\in\mathbb{C}$ with $|c_{1}|\leqslant 2$ and optimize $\mathop{\mathtt{Re}}H$ in the disk described by the second of the inequalities in (3.4). The maximum is achieved for ${c_{2}=c^{*}_{2}:=(\mathop{\mathtt{Re}}c_{1})^{2}+i\,\mathop{\mathtt{Re}}c_{1}\,\mathop{\mathtt{Im}}c_{1}-2}$. For this value of $c_{2}$ and taking into account (3.9), we get

[TABLE]

where $a=:a^{\prime}+ia^{\prime\prime}$, $c_{1}=:c_{1}^{\prime}+ic_{1}^{\prime\prime}$, and $C$ is a quantity independent of $c_{1}$. The absolute minimum of (3.10) is achieved at c_{1}^{\star}:=e^{t}\big{(}-a^{\prime}/(1+k)+ia^{\prime\prime}/(1-k)\big{)}. Moreover, even if $|c_{1}^{\star}|>2$, the minimum point $c_{1}^{*}$ of (3.10) over the disk $|c_{1}|\leqslant 2$ still satisfies

[TABLE]

where $\mathop{\mathrm{sgn}}x:=x/|x|$ for $x\in\mathbb{R}\setminus\{0\}$ and $\mathop{\mathrm{sgn}}0:=0$. For the optimal trajectory, according to (3.2), we have

[TABLE]

which would contradict (3.11) whenever $a^{\prime\prime}\neq 0$. Therefore, $a$ is real and

[TABLE]

Consider two cases. First suppose that $a=0$. Then $c_{1}^{*}(t)=0$ and $c_{2}^{*}(t)=-2$ for all $t\geqslant 0$. Note that for $(c_{1},c_{2})=(c_{1}^{*},c_{2}^{*})$ in (3.4), the first condition is satisfied with the strict inequality sign, while in the second condition equality occurs. Therefore, see, e.g., [43, Theorem IV​. 23],

[TABLE]

for some constants $\lambda\in(0,1)$ and $\mu_{1}\neq\mu_{2}$ on the unit circle (possibly depending on $t$). Comparing the coefficients of $z$ and $z^{2}$ with $c_{1}^{*}$ and $c_{2}^{*}$, we conclude that ${\lambda=1/2}$, ${\mu_{1,2}=\pm i}$, and hence $p(z,t)=(1-kz^{2})/(1+kz^{2})$. The corresponding function $f\in\mathcal{S}_{k}^{B}$ is $f(z)=f_{1}(z):=z/(1-kz^{2})$, with $a_{3}|_{f=f_{1}}=k$.

Now suppose that $a\neq 0$. Denote t_{0}:=\max\big{\{}0,\,\log|2(1+k)/a|\big{\}}. Then according to (3.13), $c_{1}^{*}(t)=-e^{t}a/(1+k)$ whenever ${0\leqslant t<t_{0}}$, and $c_{1}^{*}(t)=-2\mathop{\mathrm{sgn}}a$ for all $t\geqslant t_{0}$. Substituting $c_{1}(t):=c_{1}^{*}(t)$ into (3.12), we get

[TABLE]

It follows that $t_{0}=(1-k)/(2k)$ and

[TABLE]

Using (3.2) and (3.3), we obtain

[TABLE]

where $p_{2}^{*}(t)$ is the value of $p_{2}$ that corresponds to \big{(}c_{1},c_{2}\big{)}=\big{(}c_{1}^{*}(t),c_{2}^{*}(t)\big{)}. Elementary calculations yield p_{2}^{*}(t)=2k\big{(}e^{2(t-t_{0})}(1+k)-1\big{)} when ${0\leqslant t\leqslant t_{0}}$, $p_{2}^{*}(t)=2k^{2}$ for all ${t\geqslant t_{0}}$, and hence

[TABLE]

This gives the maximal value of $\mathop{\mathtt{Re}}a_{3}$ (and hence of $|a_{3}|$) in $\mathcal{S}_{k}^{B}$. There are two extremal functions for $\mathop{\mathtt{Re}}a_{3}$, which we denote by $f_{\pm}$, corresponding to two possible choices of the sign in (3.14). Since $z\mapsto-f(-z)$ has the same coefficient $a_{3}$ as $f$, it is clear that $f_{-}(z)=-f_{+}(-z)$, and the set of all extremal functions for $|a_{3}|$ coincides with the rotations of $f_{+}$. Therefore, we may assume the sign “$+$” in (3.14). Then the same method as in case $a=0$ allows us to write down the corresponding Heglotz function explicitly,

[TABLE]

Unfortunately, it does not seem possible to get an explicit formula for the extremal function $f_{+}$ and the Loewner chain generated by the above Herglotz function. However, one can find the Beltrami coefficient of the Becker extension provided by this Loewner chain, see e.g. [16, Proof of Theorem 2],

[TABLE]

where \rho(k):=e^{t_{0}}=\exp\big{(}(1/k-1)/2\big{)}. ∎

Proof of Theorem 3.2.

Note that $|a_{3}|\leqslant|a_{3}-\alpha a_{2}^{2}|+\alpha|a_{2}|^{2}$ for any $\alpha\in(0,1)$. The inequality $\max_{\mathcal{S}_{k}}|a_{3}|\leqslant\varrho(k)$ follows therefore from the Fekete – Szegő Theorem, see e.g. [13, p. 104], the well-known estimate $|a_{2}|\leqslant 2$ for the class $\mathcal{S}$, and Lehto’s Majorant Principle [31].

To show that the maximum of $|a_{3}|$ in $\mathcal{S}_{k}$ is strictly greater than in $\mathcal{S}_{k}^{B}$, fix $k\in(0,1)$ and note that for any non-constant holomorphic functional $\Phi:\mathcal{S}\to\mathbb{C}$, according to Lehto’s Majorant Principle, the function $q\mapsto\max_{\mathcal{S}_{q}}|\Phi|$ is strictly increasing. It follows that the extremal functions in the problem $|\Phi|\to\max_{\mathcal{S}_{k}}$ do not belong to $\mathcal{S}_{q}$ whenever ${q<k}$. Therefore, to complete the prove, it would be sufficient to show that the Becker q.c.-extension of the function $f_{+}$ from Theorem 3.1 whose Beltrami coefficient is given by (3.15) is not extremal, i.e. that $f_{+}$ admits a $q$-q.c. extension to $\mathbb{C}$ with some $q\in(0,k)$.

Suppose on the contrary that the above mentioned Becker extension of $f_{+}$ is extremal. Then it would satisfy the Hamilton – Krushkal condition [18, Theorem 1], see also [24, 19], which can be formulated as ~{}\sup_{\varphi}\big{|}\Lambda(\varphi)\big{|}=1, where

[TABLE]

$\mu$ is given by (3.15), and the supremum is taken over all holomorphic differentials $\varphi(z)\mathrm{d}z^{2}$ in $\Delta$ with $\|\varphi\|:=\iint_{\Delta}|\varphi(z)|\mathrm{d}x\mathrm{d}y\,\leqslant\,1$. Note that $\varphi(z)\mathrm{d}z^{2}$ does not have to be holomorphic at $\infty$, because the q.c.-extensions of $f_{+}$ that we consider are required to fix $\infty$.

The results of [19, §3] can be extended without any trouble from $\mathbb{D}$ to $\Delta$. In particular, by [19, Proposition 3.2], either \big{|}\Lambda(\varphi_{*})\big{|}=1 for some $\varphi_{*}$ with $\|\varphi_{*}\|=1$ or \big{|}\Lambda(\varphi_{n})\big{|}\to 1 as $n\to+\infty$ for some sequence $(\varphi_{n})$ with $\|\varphi_{n}\|\leqslant 1$ converging locally uniformly in $\Delta$ to zero. On the one hand, the former possibility does not hold in our case, because $\mu(z)$ is not of the form $k\overline{\phi(z)}/|\phi(z)|$, where $\phi$ is holomorphic, see [19, p. 161]. On the other hand, in terms of the Laurent development $\varphi_{n}(z)=\sum_{m=3}^{+\infty}c_{n,m}z^{-m}$, we have

[TABLE]

because for a fixed r\in\big{(}1,\rho(k)\big{)} the Cauchy estimates give $|c_{n,m}|\leqslant r^{m}\max_{|z|=r}|\varphi_{n}(z)|$. We obtained a contradiction, which shows that $f_{+}$ has a $q$-q.c. extension to $\mathbb{C}$ with ${q\in(0,k)}$, and hence the proof is complete. ∎

4. Extremal Becker extensions

Recall that a q.c.-extension $F:\mathbb{C}\to\mathbb{C}$ of a function $f\in\mathcal{S}$ is called extremal, if for any q.c.-extension $G:\mathbb{C}\to\mathbb{C}$ of $f$ we have $\mathop{\mathrm{ess\,sup}}_{|z|>1}|\mu_{G}(z)|\geqslant\mathop{\mathrm{ess\,sup}}_{|z|>1}|\mu_{F}(z)|$, where $\mu_{G}$ and $\mu_{F}$ stand for the Beltrami coefficients of $G$ and $F$, respectively. If the equality occurs in the above inequality only for $G=F$, then $F$ is said to be the uniquely extremal q.c.-extension of $f$ to $\mathbb{C}$.

There is a simple sufficient condition for a q.c.-extension $F:\mathbb{C}\to\mathbb{C}$ to be uniquely extremal. A (regular) Teichmüller mapping of a domain $D$ is a q.c-mapping $F:D\to\overline{\mathbb{C}}$ such that $\mu_{F}(z)=k\overline{\varphi(z)}/{|\varphi(z)|}$ for a.e. $z\in D$, where $k\in(0,1)$ and ${\varphi(z)\,\mathrm{d}z^{2}}$, ${\varphi\not\equiv 0}$, is a holomorphic quadratic differential in $D$. It is known that [41, Theorem 4] if a q.c.-extension of $f\in\mathcal{S}$ to $\mathbb{C}$ is Teichmüller on $\Delta:=\mathbb{C}\setminus\overline{\mathbb{D}}$ with $\varphi$ satisfying $\|\varphi\|:={\iint_{\Delta}|\varphi(z)|\,\mathrm{d}x\mathrm{d}y<+\infty}$, then $F$ is uniquely extremal.

Remark 4.1*.*

If $\varphi$ is holomorphic in $\Delta$ and has a zero of order at least four at $\infty$, then a q.c.-map of $\Delta$ with the Beltrami coefficient $k\overline{\varphi}/|\varphi|$ is a Teichmüller mapping of the simply connected domain $\overline{\mathbb{C}}\setminus\overline{\mathbb{D}}$. For this case, certain conditions weaker than $\|\varphi\|<+\infty$ are sufficient for (unique) extremality, see e.g. [23, 44, 45] and references therein.

Using the above mentioned sufficient condition, we construct a quite large family of functions $f\in\mathcal{S}_{k}$ with uniquely extremal extensions obtained via Becker’s construction. The idea comes from the following example. Consider the function ${f_{\sigma}\in\mathcal{S}}$, ${\sigma\in(0,2)}$, obtained by composing $\mathbb{H}:=\{z\colon\mathop{\mathtt{Re}}z>0\}\ni\zeta\mapsto\zeta^{\sigma}$, $1\mapsto 1$, with suitable Moebius transformations. This function admits a unique $|\sigma-1|$ - q.c. extension $F_{\sigma}:\mathbb{C}\to\mathbb{C}$ and belongs to $\mathrlap{\hskip 1.25pt\widetilde{\hphantom{\mathcal{S}}\vphantom{\vbox to6.45831pt{\vss}}}}\mathcal{S}_{|\sigma-1|}^{B}\vphantom{\vbox to9.90276pt{\vss}}$, see [16, Example 2]. The Beltrami coefficient of $F_{\sigma}$ is $\mu(z)={(\sigma-1)\overline{\varphi(z)}/|\varphi(z)|}$, ${\varphi(z):=1/(z^{2}-1)^{2}}$, for all $z\in\Delta$, which can be written as $\mu(\rho\zeta)=\zeta^{2}\,\psi_{\rho}(\zeta)$ for all $\rho>1$ and $\zeta\in\partial\mathbb{D}$, where $\psi_{\rho}(\zeta):=(\sigma-1)(\zeta^{2}-1/\rho^{2})/(1-\zeta^{2}/\rho^{2})$. The latter means that $F$ is Becker’s q.c.-extension (2.3) with the Herglotz function p(z,t):=\big{(}1-\psi_{e^{t}}(z)\big{)}/\big{(}1+\psi_{e^{t}}(z)\big{)}. Note that, up to the factor $(\sigma-1)$, $\psi_{\rho}$ is a Blaschke product. It turns out that any finite Blaschke product gives rise to a similar example.

Proposition 4.2**.**

Let $k\in(0,1)$, $n\in\mathbb{N}$, $a_{1},\ldots,a_{n}\in\mathbb{D}$, $\alpha\in\mathbb{R}$. Then the Herglotz function

[TABLE]

satisfies Becker’s condition (2.2) and formula (2.3) defines a uniquely extremal $k$-q.c. extension of the function $f\in\mathcal{S}$ generated by $p$. In particular, $f\in\mathrlap{\hskip 1.25pt\widetilde{\hphantom{\mathcal{S}}\vphantom{\vbox to6.45831pt{\vss}}}}\mathcal{S}_{k}^{B}\vphantom{\vbox to9.90276pt{\vss}}\setminus\mathcal{S}_{k}^{B}$ if $a_{k}\neq 0$ for all $k=1,\ldots,n$; otherwise, $f\in\mathcal{S}_{k}^{B}$.

Proof.

Condition (2.2) holds trivially because for all $t\geqslant 0$, $\psi_{t}$ is a Blaschke product. We can find the Beltrami coefficient of the $k$-q.c. extension $F$ given by (2.3), see e.g. [16, §4],

[TABLE]

for all $\rho>1$ and $\zeta\in\partial\mathbb{D}$. Hence $F|_{\mathbb{C}\setminus\overline{\mathbb{D}}}$ is a Techmüller mapping. Moreover, it is easy to see that $\|\varphi\|<+\infty$. Therefore, $F$ is the uniquely extremal q.c.-extension of $f$ to $\mathbb{C}$.

To complete the proof it remains to notice that the normalization ${p(0,t)=1}$ for a.e. ${t\geqslant 0}$ holds only if at least one of the points $a_{k}$ coincides with the origin. ∎

Remark 4.3*.*

Recently, using the generalization of Becker’s construction due to Betker [8], Sugawa [42] established a sufficient condition for a Beltrami coefficient in $\mathbb{D}$ to be trivial, i.e. to be the Beltrami coefficient of some q.c.-automorphism of $\mathbb{D}$ whose continuous extension to $\overline{\mathbb{D}}$ coincides on $\partial\mathbb{D}$ with the identity map. There is a natural one-to-one correspondence between Beltrami coefficients $\nu\in L^{\infty}(\mathbb{D})$ satisfying Sugawa’s condition and Becker’s q.c.-extensions. In particular, the $k$-q.c. extension of $f$ defined in Proposition 4.2 corresponds to $\nu(e^{-t}\zeta)=k\zeta^{2}\psi_{t}(\zeta)=k\,\phi(e^{-t}\zeta)/|\phi(e^{-t}\zeta)|$ for all $t>0$ and $\zeta\in\partial\mathbb{D}$, where  \phi(z):=e^{i\alpha}z^{n+2}\big{/}\prod_{j=1}^{n}(1-\overline{a}_{j}z)^{2}, $z\in\mathbb{D}$. This resembles Teichmüller mappings except that $\phi(z)$ in the numerator does not carry conjugation.

5. Relation between classes $\mathcal{S}_{k}$ and $\mathcal{S}_{k}^{B}$

Although $\mathcal{S}_{k}^{B}$ represents only a part of $\mathcal{S}_{k}$, see e.g. [16, §5], it is plausible to believe that Becker extendible mappings should have yet undiscovered but essential role for the study of conformal mappings admitting quasiconformal extensions.

First of all, functions of the form $f_{n}(z):=z/(1-ke^{-i\theta}z^{n})^{2/n}$, $n\in\mathbb{N}$, $\theta\in\mathbb{R}$, seem to play an important role in extremal problems for $\mathcal{S}_{k}$, similar to that of the Koebe function $f(z):=z/(1-z)^{2}$ for the whole class $\mathcal{S}$. In fact, $f_{1}$ and $f_{2}$ are to known to be extremal in some classical problems, see e.g. [27, 30]. It is not difficult to see that $f_{n}\in\mathcal{S}_{k}^{B}$ for all $n\in\mathbb{N}$. Moreover, according to Proposition 4.2, there is an infinite family of functions $f\in\mathcal{S}_{k}$ for which the uniquely extremal quasiconformal extension to $\mathbb{C}$ is a Becker extension and hence f\in\mathcal{S}_{k}^{B}\,\big{\backslash}\bigcup_{0<\nu<k}\,\mathcal{S}_{\nu}.

Secondly, there exists $k_{*}\in(0,1]$ such that for any $k\in(0,k_{*})$ we have $\mathcal{S}_{k}\subset\mathcal{S}_{q}^{B}$ with some $q\in(0,1)$ depending only on $k$. In fact, it is easy to see that $k_{*}\geqslant 1/6$. Indeed, on the one hand, $|f^{\prime\prime}(z)/f^{\prime}(z)|\leqslant 6(1-|z|^{2})$ for all $z\in\mathbb{D}$ and any $f\in\mathcal{S}$, see e.g. [14, Ch. II, §4, ineq. (6)], with $6$ replaced by $6k$ if $f\in\mathcal{S}_{k}$ thanks to Lehto’s Majorant Principle, see e.g. [29, §22]. On the other hand, if a holomorphic function $f:\mathbb{D}\to\mathbb{C}$ satisfies $|f^{\prime\prime}(z)/f^{\prime}(z)|\leqslant k(1-|z|^{2})$ for all $z\in\mathbb{D}$, then $f\in\mathcal{S}_{k}^{B}$, see [3, Satz 4.1].

We are able to improve slightly the estimate $k_{*}\geqslant 1/6$, see Corollary 6.7. In this connection, it is natural to put forward the following problem.

Problem 1**.**

Find $k_{*}$. In particular, is it true that $k_{*}=1$, i.e. that for any $k\in(0,1)$ there exists $q\in(0,1)$ such that $\mathcal{S}_{k}\subset\mathcal{S}_{q}^{B}$?

It seems interesting to consider also a bit weaker version of the latter question.

Problem 2**.**

Is it true that for any function $f\in\mathcal{S}$ admitting a q.c.-extension to $\mathbb{C}$, there exists $q\in(0,1)$, possibly depending on $f$, such that $f\in\mathcal{S}_{q}^{B}$?

Note that it is possible to replace $\mathcal{S}_{k}^{B}$ with $\mathrlap{\hskip 1.25pt\widetilde{\hphantom{\mathcal{S}}\vphantom{\vbox to6.45831pt{\vss}}}}\mathcal{S}_{k}^{B}\vphantom{\vbox to9.90276pt{\vss}}$ in the above problems as the following proposition shows.

Proposition 5.1**.**

For any $k\in(0,1)$, $\mathrlap{\hskip 1.25pt\widetilde{\hphantom{\mathcal{S}}\vphantom{\vbox to6.45831pt{\vss}}}}\mathcal{S}_{k}^{B}\vphantom{\vbox to9.90276pt{\vss}}\subset\mathcal{S}^{B}_{\kappa(k)}$, where $\kappa(k):=2k/(1+k^{2})$.

Proof.

If a function $f\in\mathrlap{\hskip 1.25pt\widetilde{\hphantom{\mathcal{S}}\vphantom{\vbox to6.45831pt{\vss}}}}\mathcal{S}_{k}^{B}\vphantom{\vbox to9.90276pt{\vss}}$ is generated by a Herglotz function $p$ satisfying (2.2), then the identity

[TABLE]

and $Q(t):=\int_{0}^{t}\!p(0,s)\mathrm{d}s$, defines a Herglotz function $p_{0}$ that obeys the normalization $p_{0}(0,t)=1$ for a.e. $t\geqslant 0$ and, moreover, generates the same function $f$. The latter can be verified using the change of variables $\tau:=Q(t)$, $\omega(\tau):=e^{i\mathop{\mathtt{Im}}Q(t)}w(t)$ that transforms the Loewner – Kufarev ODE (1.1) to $\mathrm{d}\omega/\mathrm{d}\tau=-\omega p_{0}(\omega,\tau)$.

Note that $L^{\!\mathbb{D}}_{t}:=H\circ L_{t}\circ H^{-1}$, where $H(\zeta):=(\zeta-1)/(\zeta+1)$, is an automorphism of $\mathbb{D}$ that sends z_{0}(t):=H\big{(}p_{t}(0,t)\big{)} to [math]. Taking into account that by (2.2), H\big{(}p(\mathbb{D},t)\big{)}\subset k\overline{\mathbb{D}} for a.e. $t\geqslant 0$, we see that H\big{(}p_{0}(\mathbb{D},t)\big{)}=L_{t}^{\!\mathbb{D}}\big{(}H\big{(}p(\mathbb{D},t)\big{)}\big{)} is contained for a.e. $t\geqslant 0$ in $\kappa\overline{\mathbb{D}}$, where $\kappa:=2k/(1+k^{2})$. The conclusion of the proposition follows immediately. ∎

One natural way to attack the above Problems 1 and 2 would be to propose several constructions of Loewner chains $(f_{t})$ starting from an arbitrary given function ${f_{0}=f\in\mathcal{S}_{k}}$, with images $f_{t}(\mathbb{D})$ being Jordan domains for all $t\geqslant 0$, and try to find out whether the map $F:\mathbb{C}\to\mathbb{C}$ defined by (2.3) is quasiconformal for any of these constructions.

Here we examine two quite natural constructions and show that unfortunately, both fail in general. Fix some locally absolutely continuous function $\omega:[0,+\infty)\to\partial\mathbb{D}$ and let ${\rho:[0,+\infty)\to[1,+\infty)}$ be a strictly increasing continuous function with ${\rho(0)=1}$ and ${\lim_{t\to+\infty}\rho(t)=+\infty}$.

Construction 1. Let $\Phi:\mathbb{C}\to\mathbb{C}$ be a $k$-q.c. map such that $f_{0}:=\Phi|_{\mathbb{D}}\in\mathcal{S}$. For ${t\geqslant 0}$, denote by $f^{\Phi}_{t}$, the conformal map of $\mathbb{D}$ onto \Phi\big{(}\rho(t)\mathbb{D}\big{)} normalized by ${f^{\Phi}_{t}(0)=0}$, ${\omega(t)(f^{\Phi}_{t})^{\prime}(0)>0}$. For a suitable choice of the function $\rho$, the family $(f^{\Phi}_{t})_{t\geqslant 0}$ is a Loewner chain. Using Courant’s Theorem, see e.g. [43, Theorem IX.14], it is possible to show that formula (2.3) defines a homeomorphism $F$ of $\mathbb{C}$. Construction 2. Let $f\in\mathcal{S}_{k}$. Denote by $g$ the conformal map of $\mathbb{C}\setminus\overline{\mathbb{D}}$ onto $\mathbb{C}\setminus\overline{f(\mathbb{D})}$. For ${t\geqslant 0}$, consider the conformal map $f_{t}^{g}$ of $\mathbb{D}$, ${f^{g}_{t}(0)=0}$, {\omega(t)\big{(}f^{g}_{t}\big{)}^{\prime}(0)>0}, onto the Jordan domain bounded by {g\big{(}\{z:|z|=\rho(t)\}\big{)}}. For a suitable choice of the function $\rho$, the family $(f^{g}_{t})_{t\geqslant 0}$ is a Loewner chain and the map $F$ that it generates via (2.3) is a homeomorphism of $\mathbb{C}$. We will say that the function $\rho$ is admissible in Construction 1 or, respectively, in Construction 2, if the family $(f_{t}^{\Phi})$, or respectively, the family $(f^{g}_{t})$ is a Loewner chain. Note that admissibility of $\rho$ does not depend on the choice of $\omega$.

Proposition 5.2**.**

There exists a $(1/\sqrt{2})$-q.c. map $\Phi:\mathbb{C}\to\mathbb{C}$ with $f:=\Phi|_{\mathbb{D}}\in\mathcal{S}$ such that the homemorphisms $F$ defined in Constructions 1 and 2 are not quasiconformal for any admissible ${\rho:[0,+\infty)\to[1,+\infty)}$ and any locally absolutely continuous ${\omega:[0,+\infty)\to\partial\mathbb{D}}$.

Proof.

Consider the function

[TABLE]

choosing the unique single-valued branch in $\mathbb{D}$ that belongs to $\mathcal{S}$. It is not difficult to check that

[TABLE]

In particular, \big{|}\arg p(z)\big{|}\leqslant\pi/4. Therefore, by a result of Betker [8, p. 110], see also [22, §5.1], $f$ can be extended to a $(1/\sqrt{2})$ - q.c. automorphism $\Phi:\mathbb{C}\to\mathbb{C}$ as follows.

The image $f(\mathbb{D})$ is a starlike Jordan domain symmetric w.r.t. $\mathbb{R}$ and bounded by two segments of logarithmic spirals. Namely, $\partial f(\mathbb{D})={\{2\exp(-\pi/2+|\theta|+i\theta)\colon\theta\in[-\pi,\pi]\}}$. It follows that for any $z\in\mathbb{D}\setminus\{0\}$ the intersection of $\partial f(\mathbb{D})$ and $\{tf(z):t\geqslant 0\}$ consists of one point $\zeta(z)$, with r(z):=|\zeta(z)|=2\exp\!\big{(}|\!\mathop{\operator@font Arg}\nolimits f(z)|-\pi/2\big{)}, where $\mathop{\operator@font Arg}\nolimits w$ stands for the value of $\arg w$ that belongs to $(-\pi,\pi]$.

Betker’s q.c.-extension of $f$, see e.g. [22, eq. (5.6) with $\lambda:=0$], is given by

[TABLE]

for all $z\in\mathbb{C}\setminus\mathbb{D}$, where $\eta(z):=\mathop{\mathrm{sgn}}\mathop{\mathtt{Im}}z$. Simple calculations give

[TABLE]

Using the above formulas we see that for any $r>1$ the boundary of $D_{r}:=\Phi(r\mathbb{D})$ consists of two real-analytic arcs with common end-points at $\Phi(\pm r)$, where they form angle of magnitude $2\mathop{\operator@font arctg}\nolimits(1/2)<\pi/2$. The angle at $\Phi(-r)$ is internal w.r.t. $D_{r}$. It follows that conformal mappings of $\mathbb{D}$ onto $D_{r}$ do not belong to the Hardy space $H^{2}(\mathbb{D})$. Therefore, by the main result of [6], there is no Loewner chain with image domains $D_{r}$ that defines a q.c.-extension via (2.3). Therefore, the homeomorphism $F$ in Construction 1 generated by the Loewner chain $(f^{\Phi}_{t})$ is not quasiconformal, whichever $\rho$ and $\theta$ we choose.

\stackMath

Let us now consider Construction 2 with the same function ${f\in\mathcal{S}_{1/\sqrt{2}}}$ as above. One remarkable property of $\Omega:=f(\mathbb{D})$ is that $\{1/z\colon z\in\overline{\mathbb{C}}\setminus\overline{\Omega}\}=-\tfrac{1}{4}\Omega.$ It follows that, up to rotation, $g(z)=-4/f(-1/z)$ for all $z\in\mathbb{C}\setminus\overline{\mathbb{D}}$. Suppose that for a suitable choice of the functions $\rho$ and $\omega$, the homeomorphism $F:\mathbb{C}\to\mathbb{C}$ defined with the help of the Loewner chain $(f^{g}_{t})$ is $k$-quasiconformal for some ${k\in(0,1)}$. Then arguing as in [16, proof of Theorem 2], we see that for all ${t\geqslant 0}$ aside from some null-set $N$, $\vphantom{f_{t}}\smash{\stackengine{-.075ex}{f_{t}}{\boldsymbol{\cdot\hskip 1.75003pt}}{O}{r}{F}{\useanchorwidth}{S}}:=\partial f_{t}/\partial t$ and $f_{t}^{\prime}$ exist a.e. on $\partial\mathbb{D}$, do not vanish, and

[TABLE]

where $U(k)$ is defined in Theorem A. Moreover, by construction, $\partial f_{t}(\mathbb{D})$ is $C^{\infty}$ when ${t>0}$. Hence, in fact, $f_{t}^{\prime}$ extends smoothly to $\partial\mathbb{D}$ for all $t>0$; see e.g. [36, Chapter 3].

Taking into account that \big{|}g^{-1}\big{(}f_{t}(e^{i\theta})\big{)}\big{|}=\rho(t) for all ${t\geqslant 0}$ and all ${\theta\in[0,2\pi]}$, it follows that $\rho^{\prime}(t)$ exists for any $t\in(0,+\infty)\setminus N$ and for the normal velocity of $\partial f_{t}(\mathbb{D})$ we have

[TABLE]

Together with (5.3) this implies that on the one hand, for any $t\in(0,+\infty)\setminus N$,

[TABLE]

On the other hand,

[TABLE]

Combining (5.4) with (5.5), we see that

[TABLE]

Therefore, the conformal weldings \gamma_{t}:=\big{(}g^{-1}\circ f_{t}|_{\partial\mathbb{D}}\big{)}\big{/}\rho(t), $t\in(0,+\infty)$, are $K^{2}$-Lipschitz continuous. Using Carathéodory’s Extension Theorem (see e.g. [36, p. 18]) and Courant’s Theorem (see e.g. [43, Theorem IX.14]) we conclude that $\gamma_{t}\to\gamma_{0}$ as $t\to 0^{+}$. It follows that $\gamma_{0}$ has to be also Lipschitz-continuous, but in reality it is not. This contradiction shows that $F$ is not quasiconformal. ∎

6. A sufficient condition for Becker extendibility

Below we prove a sufficient condition for a holomorphic function to be Becker extendible, i.e. to have a q.c.-extension of the form (2.3). This simple result is probably known to specialists: somewhat similar ideas appeared e.g. in [7] and [20, equation (11)]. However, it does not seem to be ever stated in the form as presented below. For the notions of a meromorphic function of several complex variables and that of an analytic set we refer the reader to [39, §15, §8].

Theorem 6.1**.**

Let $f$ be a holomorphic function in $\mathbb{D}$, with $f^{\prime}(0)-1=f(0)=0$. Suppose that there exists a meromorphic solution $\Phi:\mathbb{C}\times\mathbb{D}\to\mathbb{C}$ to the PDE initial value problem

[TABLE]

with a coefficient $\varphi$ meromorphic in $\mathbb{C}\times\mathbb{D}$ and satisfying the following two conditions:

• (i)

$\varphi(0,0)=0$;

• (ii)

$r\!\left|\varphi(w/r,w)\right|\leqslant k$* for all $w\in\mathbb{D}$ and all r\in\big{(}|w|^{2},1\big{)}.*

Suppose also that there exists $\varepsilon\in(0,1)$ and $M>0$ such that

[TABLE]

Then $f$ admits a $k$-q.c. Becker extension given by

[TABLE]

In particular, $f\in\mathcal{S}_{k}^{B}$.

Remark 6.2*.*

Since $\Phi(0,0)=f(0)=0$, it is sufficient to check condition (6.3) only for $z$ large enough.

Before proving Theorem 6.1, let us consider a few examples.

Example 6.3**.**

Let $f$ be a holomorphic function in $\mathbb{D}$ with $f^{\prime}(0)-1=f(0)=0$. Set $\varphi(z,w):=(z-w)f^{\prime\prime}(w)/f^{\prime}(w)$. Then $\Phi(z,w):=f(w)+(z-w)f^{\prime}(w)$ solves problem (6.1), (6.2) and satisfies (6.3). Condition (i) in Theorem 6.1 holds trivially, while (ii) is equivalent to $(1-|w|^{2})|wf^{\prime\prime}(w)/f^{\prime}(w)|\leqslant k$, which is a classical sufficient condition for q.c.-extendibility.

Example 6.4**.**

Similarly, setting $\varphi(z,w):=f^{\prime}(w)-1$ and $\Phi(z,w):=f(w)+z-w$, we recover another well-known sufficient condition for q.c.-extendibility \big{|}f^{\prime}(w)-1\big{|}\leqslant k, $w\in\mathbb{D}$, see [12, §3].

The following corollary represents another example.

Corollary 6.5**.**

Fix $k\in(0,1)$. Let $f(z)=z+a_{2}z^{2}+\ldots$ be holomorphic in $\mathbb{D}$. If

[TABLE]

then $f\in\mathcal{S}_{k}^{B}$, with its Becker extension given by $F(z)=\Phi(z,1/\bar{z})$ for all $z\in\mathbb{C}\setminus\overline{\mathbb{D}}$, where

[TABLE]

Proof.

Let \varphi(z,w):=2a_{2}(z-w)+(z-w)^{2}\big{(}a_{2}^{2}+\tfrac{1}{2}S_{f}(w)\big{)}, where $S_{f}$ stands for the Schwarzian derivative of $f$. Then $\Phi$ given by (6.6) solves problem (6.1), (6.2).

Moreover, there exists $K>1$ such that \big{|}f^{\prime}(w)\big{|}\leqslant K and \left|a_{2}-\frac{1}{2}\big{(}{f^{\prime\prime}(w)}/{f^{\prime}(w)}\big{)}\right|\leqslant K|w| for all $w\in\frac{1}{2}\mathbb{D}$. Hence, for any $(z,w)\in\mathbb{C}\times\mathbb{D}$ with $|w|\leqslant|z|$ and ${|zw|\leqslant\varepsilon^{2}}:=(4K)^{-1}$,

[TABLE]

This proves (6.3). Finally, since $|w|(1-|w|^{2})\leqslant 2\sqrt{3}/9$ for all $w\in\mathbb{D}$, condition (6.5) ensures that $\varphi$ satisfies (ii), while (i) holds trivially.

Thus, the desired conclusion takes place due to Theorem 6.1. ∎

Remark 6.6*.*

A well-known result by Ahlfors and Weill [2], see also [1], asserts that if a holomorphic function $f:\mathbb{D}\to\mathbb{C}$ satisfies $\frac{1}{2}(1-|z|^{2})^{2}|S_{f}(z)|\leqslant k$, where $k\in(0,1)$, for all $z\in\mathbb{D}$, then $f$ is univalent and extends to a $k$-q.c. automorphism $F:\overline{\mathbb{C}}\to\overline{\mathbb{C}}$, with $F$ given by an explicit formula. This extension can be obtained with the help of Becker’s construction (see, e.g., [3, §4] and [5]), but it does not have to fix $\infty$ and hence $F$ is not a Becker extension in general (which was overlooked in [16, §5]). Corollary 6.5 is a sort of modification of the Ahfors – Weill condition that ensures extendibility to a q.c.-automorphism of $\mathbb{C}$. In fact, if $a_{2}=0$ then the q.c.-extension of $f$ given in Corollary 6.5 coincides with the extension constructed by Ahlfors and Weill [1].

It is known, see e.g. [29, Example 4 on p. 132], that given $k\in(0,1)$, for all $f\in\mathcal{S}_{k}$, $|a_{2}|\leqslant 2k$ and $|S_{f}(z)|\leqslant 6k/(1-|z|^{2})^{2}$ for any $z\in\mathbb{D}$. Therefore, Corollary 6.5 implies immediately the following statement.

Corollary 6.7**.**

If $0<k<0.188856\ldots$ , then $\mathcal{S}_{k}\subset\mathcal{S}_{q}^{B}$ with $q:=(3+\tfrac{8\sqrt{3}}{9})k+4k^{2}$.

It remains to prove the main result of this section.

Proof of Theorem 6.1.

Let

[TABLE]

Thanks to condition (6.3), these functions are holomorphic in $\zeta\in\varepsilon\mathbb{D}$ and real-analytic in $t\geqslant 0$, and moreover, $|f_{t}(\zeta)|\leqslant Me^{t}$ for all $\zeta\in\varepsilon\mathbb{D}$ and $t\geqslant 0$.

Note also that for any fixed $t\geqslant 0$, the point $(\zeta e^{t},\zeta e^{-t})$ can lie in the polar set $\mathcal{P}$ of $\Phi$ only for $\zeta$ belonging to a discrete subset of $\mathbb{D}$. Otherwise, since $\mathcal{P}$ is an analytic set in $\mathbb{C}\times\mathbb{D}$, we would have that $(\zeta e^{t},\zeta e^{-t})\in\mathcal{P}$ for all $\zeta\in\mathbb{D}$, which contradicts (6.3). Therefore, $p(\cdot,t)$ and $f_{t}$ are well-defined meromorphic functions in $\mathbb{D}$ for each $t\geqslant 0$.

As an elementary calculation shows, for all $\zeta\in\mathbb{D}$ and $t>0$,

[TABLE]

where $r:=e^{-2t}$ and $w:=\zeta e^{-t}$. Trivially, $|w|^{2}<r<1$. Therefore, condition (ii) implies that $p$ is a Herglotz function satisfying Becker’s condition (2.2).

Clearly, $f_{t}(0)=\Phi(0,0)=f(0)=0$, $t\geqslant 0$. Moreover, taking into account (i), we have $f^{\prime}_{t}(0)=e^{t}\Phi^{\prime}_{z}(0,0)+e^{-t}\Phi^{\prime}_{w}(0,0)=e^{t}\Phi^{\prime}_{z}(0,0)=e^{t}f^{\prime}(0)=e^{t}$ for all $t\geqslant 0$.

We see that $(f_{t})$ satisfies the hypothesis of Pommerenke’s Criterion [35, Theorem 6.1 on p. 159]. Hence $(f_{t})$ is a classical radial Loewner chain. Furthermore, by Theorem A, $f=f_{0}$ admits a $k$-q.c. extension $F:\mathbb{C}\to\mathbb{C}$ given by the formula

[TABLE]

which was to be proved. ∎

Acknowledgement

The authors are grateful to Professor Toshiyuki Sugawa for fruitful discussions on the topic of the present paper and, in particular, for drawing their attention to reference [42].