---
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 $\C$. In particular, we answer a question on estimation of $|a_3|$ raised by Kühnau 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 $\C$ with $k\le k_*$ 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 $\C$ with $k''<k$.'
address:
- 'Institutt for matematikk og fysikk, Universitetet i Stavanger, 4036 Stavanger, '
- 'Department of Applied Science, Yamaguchi University, 2-16-1 Tokiwadai, Ube 755-8611, '
author:
- Pavel Gumenyuk
- 'Ikkei Hotta$^\dag$'
title: 'Univalent functions with quasiconformal extensions: Becker’s class and estimates of the third coefficient.'
---

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. [@Lehto_book; @Schober]. Let $k\in(0,1)$. A function $f$ holomorphic in a domain $D\subset{\mathbb{C}}$ is said to be 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 $${\mathbb{D}}\ni z\mapsto f(z)=z+\sum_{n=2}^{+\infty}a_nz^n.$$ One of the main tools to study this class is the *parametric representation*, which goes back to Loewner [@Loewner], see e.g. [@Pommerenke §6.1], see also [@Kufarev43; @Pommerenke65; @Gut]. 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 $\Re p(\cdot,t)\ge0$ for a.e. $t\ge0$. It is known that for any Herglotz function $p$, the initial value problem for the Loewner–Kufarev ODE $$\label{EQ_LK-ODE0}
\frac{{\mathrm{d}}w}{{\mathrm{d}}t}=-w\,p(w,t),\quad t\ge0,\qquad w(z,0)=z\in{\mathbb{D}},$$ has a unique solution $w=w(z,t)$ and the locally uniform limit $$\label{EQ_f-limit}
f(z):=\lim_{t\to+\infty}\frac{\,\mathrlap{w(z,t)}\hphantom{w'(0,t)}}{\,w'(0,t)},\quad z\in{\mathbb{D}},$$ where $w'$ denotes the derivative w.r.t. $z$, exists and belongs to ${\mathcal{S}}$. On the other hand, see e.g. [@Pommerenke Theorem 6.1 on p.159] or [@Gut], *every* function $f\in{\mathcal{S}}$ can be represented by  with a suitable, and in general not unique, *normalized* Herglotz function, i.e. a Herglotz function $p$ with $\Re p(0,t)=1$ for a.e. $t\ge0$.

A natural problem arises: *given a subclass ${\mathrlap{\hskip.125em\widetilde{\hphantom{{\mathcal{S}}}\vphantom{\vbox to1.5ex{\vss}}}}{\mathcal{S}}\vphantom{\vbox to2.3ex{\vss}}}\subset{\mathcal{S}}$, find a class of Herglotz functions that generates ${\mathrlap{\hskip.125em\widetilde{\hphantom{{\mathcal{S}}}\vphantom{\vbox to1.5ex{\vss}}}}{\mathcal{S}}\vphantom{\vbox to2.3ex{\vss}}}$ via .* The answer is known in some cases, e.g. for starlike functions, bounded univalent functions, and for univalent functions with real Taylor coefficients;

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 [@Becker72] found a condition on $p$ in the Loewner–Kufarev equation , see Sect.\[SS\_Becker\], such that the function $f$ given by  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.\[SS\_a3\] we find the sharp estimate for $|a_3|$ in $S_k^B$, see Theorem \[TH\_a\_3\]. An immediate corollary is the answer to a question of Kühnau and Niske [@KuhnauNiske]: Theorem \[TH\_a\_3\] 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. [@Duren], with many of them being motivated by the famous Bieberbach Conjecture concerning estimates for $|a_n|$, which was proved by de Branges [@deBranges] in 1984. Unfortunately, only a few of these results have been extended to classes ${\mathcal{S}}_k$, see e.g. [@Kuhnau69; @Lehto71]. 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 \[TH\_not-extremal\].

Becker’s construction of quasiconformal extensions {#SS_Becker}
==================================================

Throughout the paper we make use of Loewner Theory, the classical version of which can be found in [@Pommerenke Chapter 6]. Following Becker [@Becker76], [@Becker80 §5.1], we replace the usual normalization ${p(0,t)=1}$ by a weaker condition $$\label{EQ_weaker-normalization-for-p}
\int_0^{+\infty}\!\!\Re p(0,t)\,{\mathrm{d}}t=+\infty,$$ which still implies that $\bigcup\limits_{t\ge0}f_t({\mathbb{D}})=\C$. In 1972, he discovered the

\[TH\_Becker\] Let $k\in[0,1)$ and let $(f_t)$ be a radial Loewner chain whose Herglotz function $p$ satisfies $$\label{EQ_Becker-condition}
  p({\mathbb{D}},t) \subset U(k):=\left\{ w \in \C \colon \left|\frac{w-1}{w+1}\right| \le k\right\}\quad\text{for  a.e.~$t\ge0$}.$$ Then for every $t\ge0$, 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 $$\label{EQ_BeckerExt}
F(\rho e^{i\theta}):=\left\{
 \begin{array}{ll}
  f_0(\rho e^{i\theta}), & \text{if~$0\le \rho<1$},\\[1ex]
  f_{\log \rho}(e^{i\theta}), & \text{if~$\rho\ge1$}.
 \end{array}
\right.$$

\[RM\_2018\] According to [@Istvan 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  is $k$-quasiconformal in $\C$, then the Herglotz function $p$ associated with $(f_t)$ satisfies Becker’s condition .

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.$t\ge0$ and satisfying . A bit larger class of all $f\in{\mathcal{S}}$ generated by Herglotz functions subject to Becker’s condition , but not necessarily normalized, will be denoted by ${\mathrlap{\hskip.125em\widetilde{\hphantom{{\mathcal{S}}}\vphantom{\vbox to1.5ex{\vss}}}}{\mathcal{S}}_{k}^B\vphantom{\vbox to2.3ex{\vss}}}$.

According to Theorem \[TH\_Becker\], ${\mathcal{S}}^B_k\subset{\mathrlap{\hskip.125em\widetilde{\hphantom{{\mathcal{S}}}\vphantom{\vbox to1.5ex{\vss}}}}{\mathcal{S}}_{k}^B\vphantom{\vbox to2.3ex{\vss}}}\subset{\mathcal{S}}_k$. It is known that ${\mathrlap{\hskip.125em\widetilde{\hphantom{{\mathcal{S}}}\vphantom{\vbox to1.5ex{\vss}}}}{\mathcal{S}}_{k}^B\vphantom{\vbox to2.3ex{\vss}}}\neq{\mathcal{S}}_k$, see . 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  appears to be sufficient for q.c.-extendibility also in the framework of the general Loewner Theory introduced in [@BracciCD:evolutionI; @BracciCD:evolutionII]; see [@Istvan], [@Ikkei2], and [@HottaGum::QC-chordal]. This discussion will be continued in Sect.\[SS\_relation\].

Estimate of the third coefficient {#SS_a3}
=================================

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 [@KuhnauNiske]: does there exist $k_0>0$ such that for any $k\in(0,k_0]$ and any function $f(z)=z+a_2z+a_3 z^3+\ldots$ belonging to ${\mathcal{S}}_k$, the inequality $|a_3|\le k$ holds?

\[TH\_a\_3\] Let $k\in(0,1)$. Then for every function $f(z)=z + a_2z^2 + a_3z^3+\ldots$ belonging to ${\mathcal{S}}_k^B$, $$|a_3|\le k\big(1+e^{1-1/k}(1+k)\big).$$ 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  of its q.c.-extension to $\C$.

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.

\[TH\_not-extremal\] For any $k\in(0,1)$, $$\label{EQ_upper-est}
\max_{{\mathcal{S}}_k^B}|a_3|<\max_{{\mathcal{S}}_k}|a_3|\le \varrho(k):=\min_{\alpha\in(0,1)}\Bigl[\big(1+2e^{-2\alpha/(1-\alpha)}\big)k+4\alpha k^2\Bigr].$$

\[RM\_Krushkal\] The sharp estimate in Theorem \[TH\_a\_3\] shows that the inequality ${|a_n|\le 2k/(n-1)}$ written in the larger class ${\mathcal{S}}_k$ for $0<k\le 1/(1+n^2)$ and all $n=2,3,\ldots$ by Krushkal [@Krushkal-a_n 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  behaves asymptotically as $4k^2$. The three estimates are shown in Figure \[FI\_graph\].

![Estimates for $|a_3|$ mentioned in Remark \[RM\_Krushkal\].[]{data-label="FI_graph"}](graphM.pdf){width="9.5cm"}

2ex

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. $t\ge0$, where $U(k)$ is the closed disk defined in Theorem \[TH\_Becker\]. As it follows from the very definition, ${\mathcal{S}}_k^B$ coincides with the image of the map $${\mathcal{H}}_k\ni p~\mapsto~f:=\lim_{t\to+\infty}e^{t}w(z,t)~\in\,{\mathcal{S}},$$ 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 . Write $p(z,t)=1+p_1(t)z+p_2(t)z^2+\ldots$ for all $z\in{\mathbb{D}}$ and a.e. $t\ge0$ and let $$f(z,t):=e^{t}w(z,t)=z+a_2(t)z^2+a_3(t)z^3+\ldots$$ 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. $t\ge0$, where $p_0(z,t)=1+c_1(t)z+c_2(t)z^2+\ldots$ is an arbitrary normalized Herglotz function and $$L(z):=\frac{1+Kz}{K+z},\quad K:=\frac{1+k}{1-k},$$ is a conformal map of ${\mathbb{H}}:=\{z\colon\Re z>0\}$ onto $U(k)$ with $L(1)=1$.

As usual, from  we obtain the initial value problem for the coefficients $a_2$ and $a_3$, $$\begin{aligned}
\frac{{\mathrm{d}}a_2}{{\mathrm{d}}t}&=-e^{-t} p_1(t)=-ke^{-t} c_1(t), & a_2(0)=0,\label{EQ_phase1}\\[.7ex]
\frac{{\mathrm{d}}a_3}{{\mathrm{d}}t}&=-e^{-2t}p_2(t)-2e^{-t}p_1(t)a_2(t)\notag\\&=-k\Big(e^{-2t}\big(c_2(t)-(1-k)\frac{c_1(t)^2}{2}\,\big)+2e^{-t}c_1(t)a_2(t)\Big), & a_3(0)=0.\label{EQ_phase2}\end{aligned}$$

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 $~\max\,\Re a_3$. The latter problem can be reformulated as the optimal control problem for the above system and the objective functional $\Re a_3(+\infty)$, with a control function ${t\mapsto \big(c_1(t),c_2(t)\big)\in\C^2}$ regarded as admissible if it is measurable and for a.e. $t\ge0$ satisfies $$\label{EQ_Cara}
 |c_1|\le 2,\qquad |2 c_2-c_1^2|\le 4-|c_1|^2.$$ Conditions  describe the value region of ${\mathcal{C}}\ni q\mapsto (c_1,c_2)\in\C^2$ 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., [@Tsuji Chapter IV, §7].

To apply Pontryagin’s Maximum Principle, we define the (holomorphic) Hamiltonian $$H(a_2,a_3,\psi_2,\psi_3,t,c_1,c_2):=-ke^{-t} c_1\psi_2-k\Big(e^{-2t}\big(c_2-(1-k)\frac{c_1^2}{2}\big)+2e^{-t}c_1a_2\Big)\psi_3$$ and write the adjoint system $$\begin{aligned}
\frac{{\mathrm{d}}\psi_2}{{\mathrm{d}}t}&=-\frac{\partial H}{\partial a_2}=2ke^{-t}c_1(t)\psi_3(t),\label{EQ_adj1}\\[1.25ex]
\frac{{\mathrm{d}}\psi_3}{{\mathrm{d}}t}&=-\frac{\partial H}{\partial a_3}=0.\label{EQ_adj2}\end{aligned}$$ The maximum of $\Re a_3(+\infty)$ is to be found among all the trajectories of , 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 [@Pontryagin 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 , supplemented with the adjoint equations , and the transversality conditions $$\label{EQ_trans}
\psi_2(+\infty)=0,\qquad \psi_3(+\infty)=1,$$ it holds that $$\begin{gathered}
\label{EQ_Max-principle}
\max_{(c_1,c_2)} \Re H\big(a_2(t),a_3(t),\psi_2(t),\psi_3(t),t,c_1,c_2\big)\\ = \Re H\big(a_2(t),a_3(t),\psi_2(t),\psi_3(t),t,c^*_1(t),c^*_2(t)\big),\end{gathered}$$ where the maximum is taken over all $(c_1,c_2)\in\C^2$ subject to conditions .

System – can be integrated using integrals to ,: $$\label{EQ_adjoint}
\psi_2(t)=a-2a_2(t),\quad \psi_3(t)=1,$$ where $a:=2a_2(+\infty)$.

To find the maximum of $\Re H$ as a function of $c_1$ and $c_2$, we first fix a $c_1\in\C$ with $|c_1|\le2$ and optimize $\Re H$ in the disk described by the second of the inequalities in . The maximum is achieved for ${c_2=c^*_2:=(\Re c_1)^2+i\,\Re c_1\,\Im c_1-2}$. For this value of $c_2$ and taking into account , we get $$\begin{aligned}
-\frac{e^{2t}}{k}\,\Re H&=e^t\,\Re(ac_1)+\frac{1+k}{2}\,c_1'^{\,2}+\frac{1-k}{2}\,c_2''^{\,2}-2\notag\\                        &=\,\frac{1+k}{2}\left(c_1'+\frac{e^ta'}{1+k}\right)^{\!\!2}\,                            +\,\frac{1-k}{2}\left(c_1''-\frac{e^ta''}{1-k}\right)^{\!\!2}\,+\,C,\label{EQ_quadratic}\end{aligned}$$ where $a=:a'+ia''$, $c_1=:c_1'+ic_1''$, and $C$ is a quantity independent of $c_1$. The absolute minimum of  is achieved at $c_1^\star:=e^t\big(-a'/(1+k)+ia''/(1-k)\big)$. Moreover, even if $|c_1^\star|>2$, the minimum point $c_1^*$ of  over the disk $|c_1|\le 2$ still satisfies $$\label{EQ_sgn}
 {\mathop{\mathrm{sgn}}}\Re c_1^*=-{\mathop{\mathrm{sgn}}}a',\qquad {\mathop{\mathrm{sgn}}}\Im c_1^*={\mathop{\mathrm{sgn}}}a'',$$ 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 , we have $$\label{EQ_a}
 a=-2k\int_0^{+\infty}\!\!e^{-t}c_1^*(t)\,{\mathrm{d}}t,$$ which would contradict  whenever $a''\neq0$. Therefore, $a$ is real and $$\label{EQ_c1}
c_1^*(t)=
\begin{cases}
-e^ta/(1+k),& \text{if~}\big|e^ta/(1+k)\big|<2,\\
-2{\mathop{\mathrm{sgn}}}a,&      \text{otherwise}.
\end{cases}$$

Consider two cases. First suppose that $a=0$. Then $c_1^*(t)=0$ and $c_2^*(t)=-2$ for all $t\ge0$. Note that for $(c_1,c_2)=(c_1^*,c_2^*)$ in , the first condition is satisfied with the strict inequality sign, while in the second condition equality occurs. Therefore, see, e.g., [@Tsuji Theorem IV.23], $$\begin{aligned}
 p_0(z,t)&=\lambda\frac{1+\mu_1 z}{1-\mu_1 z}+(1-\lambda)\frac{1+\mu_2 z}{1-\mu_2 z}\\
          &=1\,+\,2(\lambda\mu_1+(1-\lambda)\mu_2)z\,+\,2(\lambda\mu_1^2+(1-\lambda)\mu_2^2)z^2\,+\,\ldots,\quad z\in{\mathbb{D}},\end{aligned}$$ 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 $a\neq0$. Denote $t_0:=\max\big\{0,\,\log|2(1+k)/a|\big\}$. Then according to , $c_1^*(t)=-e^ta/(1+k)$ whenever ${0\le t<t_0}$, and $c_1^*(t)=-2{\mathop{\mathrm{sgn}}}a$ for all $t\ge t_0$. Substituting $c_1(t):=c_1^*(t)$ into , we get $$a=2k\left(\frac{a t_0}{1+k}+2e^{-t_0}{\mathop{\mathrm{sgn}}}a\right)=\frac{2k(1+t_0)}{1+k}a.$$ It follows that $t_0=(1-k)/(2k)$ and $$\label{EQ_a2}
a_2(+\infty)=a/2=\pm\alpha(k),\quad \text{where~$~\alpha(k):=(1+k)e^{-t_0}$.}$$ Using  and , we obtain $$a_3(+\infty)~=~a_2(+\infty)^2-\displaystyle\int\limits_0^{+\infty}\!\! e^{-2t}p^*_2(t)\,{\mathrm{d}}t,$$ 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 ${0\le t\le t_0}$, $p_2^*(t)=2k^2$ for all ${t\ge t_0}$, and hence $$a_3(+\infty)=k\big(1+e^{1-1/k}(1+k)\big)>k=a_3|_{f=f_1}.$$

This gives the maximal value of $\Re a_3$ (and hence of $|a_3|$) in ${\mathcal{S}}_k^B$. There are two extremal functions for $\Re a_3$, which we denote by $f_\pm$, corresponding to two possible choices of the sign in . 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 . Then the same method as in case $a=0$ allows us to write down the corresponding Heglotz function explicitly, $$p(z,t)=\frac{1-k z^2+(1-k)e^{t-t_0}z}{1+kz^2+(1+k)e^{t-t_0}z}~\text{~for $t\in[0,t_0]$}\quad
\text{and}\quad p(z,t)=\frac{1 - kz}{1 + kz}~\text{~for~$t\ge t_0$}.$$

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. [@Istvan Proof of Theorem 2], $$\label{EQ_extremal-Beltrami-coefficient}
\mu(z)=\frac{z^2}{|z|^2}\,\frac{p\big(z/|z|,\log |z|\big)-1}{p\big(z/|z|,\log |z|\big)+1}=
\begin{cases}
-k\dfrac{z^4}{|z|^4}\,\dfrac{\rho(k) + \bar z}{\rho(k) + z},&\text{if $|z|\in\big(1,\,\rho(k)\big)$,}\\[2.75ex]
-k\dfrac{z^3}{|z|^3}, & \text{if $|z|>\rho(k)$,}
\end{cases}$$ where $\rho(k):=e^{t_0}=\exp\big((1/k-1)/2\big)$.

Note that $|a_3|\le|a_3-\alpha a_2^2|+\alpha |a_2|^2$ for any $\alpha\in(0,1)$. The inequality $\max_{{\mathcal{S}}_k}|a_3|\le\varrho(k)$ follows therefore from the Fekete–Szegő Theorem, see e.g. [@Duren p.104], the well-known estimate $|a_2|\le 2$ for the class ${\mathcal{S}}$, and Lehto’s Majorant Principle [@Lehto_Majorant].

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 \[TH\_a\_3\] whose Beltrami coefficient is given by  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 [@Hamilton Theorem 1], see also [@Krushkal-extremal; @HarringtonOrtel], which can be formulated as $~\sup_\varphi\big|\Lambda(\varphi)\big|=1$, where $$\Lambda(\varphi):={\frac{1}{k}\iint\limits_\Delta\varphi(z)\,\mu(z)\,{\mathrm{d}}x{\mathrm{d}}y},\quad \Delta:=\{z:1<|z|<+\infty\},$$ $\mu$ is given by , and the supremum is taken over all holomorphic differentials $\varphi(z){\mathrm{d}}z^2$ in $\Delta$ with $\|\varphi\|:=\iint_\Delta|\varphi(z)|{\mathrm{d}}x{\mathrm{d}}y\,\le\,1$. 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 [@HarringtonOrtel §3] can be extended without any trouble from ${\mathbb{D}}$ to $\Delta$. In particular, by [@HarringtonOrtel Proposition 3.2], either $\big|\Lambda(\varphi_*)\big|=1$ for some $\varphi_*$ with $\|\varphi_*\|=1$ or $\big|\Lambda(\varphi_n)\big|\to1$ as $n\to+\infty$ for some sequence $(\varphi_n)$ with $\|\varphi_n\|\le1$ 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 [@HarringtonOrtel 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 $$\frac{\Lambda(\varphi_n)}{2\pi}=\frac{1+\log\rho(k)}{\rho(k)}c_{n,3}~+~\frac{\rho(k)^2-2\log\rho(k)-1}{2}~\sum_{m=4}^{+\infty}
\frac{(-1)^m c_{n,m}}{\rho(k)^{m-2}}~\text{~}\longrightarrow~0~\text{~}\text{as~}~n\to+\infty,$$ because for a fixed $r\in\big(1,\rho(k)\big)$ the Cauchy estimates give $|c_{n,m}|\le r^{m}\max_{|z|=r}|\varphi_n(z)|$. We obtained a contradiction, which shows that $f_+$ has a $q$-q.c. extension to $\C$ with ${q\in(0,k)}$, and hence the proof is complete.

Extremal Becker extensions {#SS_examples}
==========================

Recall that a q.c.-extension $F:\C\to\C$ 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 ${\mathop{\mathrm{ess\,sup}}}_{|z|>1}|\mu_G(z)|\ge{\mathop{\mathrm{ess\,sup}}}_{|z|>1}|\mu_F(z)|$, 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 $F:\C\to\C$ 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 ${\varphi(z)\,{\mathrm{d}}z^2}$, ${\varphi\not\equiv0}$, is a holomorphic quadratic differential in $D$. It is known that [@Strebel:1978 Theorem 4] if a q.c.-extension of $f\in{\mathcal{S}}$ to $\C$ is Teichmüller on $\Delta:=\C\setminus\overline{\mathbb{D}}$ with $\varphi$ satisfying $\|\varphi\|:={\iint_\Delta|\varphi(z)|\,{\mathrm{d}}x{\mathrm{d}}y<+\infty}$, then $F$ is uniquely extremal.

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. [@Huang; @YaoF; @YaoP] 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 ${\mathbb{H}}:=\{z\colon \Re z>0\}\ni \zeta\mapsto\zeta^\sigma$, $1\mapsto 1$, with suitable Moebius transformations. This function admits a unique $|\sigma-1|$-q.c. extension $F_\sigma:\C\to\C$ and belongs to ${\mathrlap{\hskip.125em\widetilde{\hphantom{{\mathcal{S}}}\vphantom{\vbox to1.5ex{\vss}}}}{\mathcal{S}}_{|\sigma-1|}^B\vphantom{\vbox to2.3ex{\vss}}}$, see [@Istvan 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  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.

\[PR\_examples\] 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 $$\label{EQ_examples_p}
  p(z,t):=\frac{1+k\psi_t(z)}{1-k\psi_t(z)},~\text{~where~}~\psi_t(z):=e^{i\alpha}\, \prod_{j=1}^n\frac{z-e^{-t}a_j}{1-e^{-t}\,\overline a_j z},\quad z\in{\mathbb{D}},~t\ge0,$$ satisfies Becker’s condition  and formula  defines a uniquely extremal $k$-q.c. extension of the function $f\in{\mathcal{S}}$ generated by $p$. In particular, $f\in{\mathrlap{\hskip.125em\widetilde{\hphantom{{\mathcal{S}}}\vphantom{\vbox to1.5ex{\vss}}}}{\mathcal{S}}_{k}^B\vphantom{\vbox to2.3ex{\vss}}}\setminus{\mathcal{S}}_k^B$ if $a_k\neq0$ for all $k=1,\ldots, n$; otherwise, $f\in{\mathcal{S}}_k^B$.

Condition  holds trivially because for all $t\ge0$, $\psi_t$ is a Blaschke product. We can find the Beltrami coefficient of the $k$-q.c. extension $F$ given by , see e.g. [@Istvan §4], $$\mu_F(\rho\zeta)=\frac{p(\zeta,\log\rho)-1}{p(\zeta,\log\rho)+1}\,\zeta^2\,= \,k\frac{\,\overline{\varphi(\rho\zeta)}\,}{|\varphi(\rho\zeta)|},
~\text{~where~} \varphi(z):={e^{-i\alpha}z^{n-2}}\,\prod_{j=1}^n\frac{1}{(z-a_j)^2},~ z\in\Delta,$$ for all $\rho>1$ and $\zeta\in\partial{\mathbb{D}}$. Hence $F|_{\C\setminus\overline{\mathbb{D}}}$ 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 $\C$.

To complete the proof it remains to notice that the normalization ${p(0,t)=1}$ for a.e. ${t\ge0}$ holds only if at least one of the points $a_k$ coincides with the origin.

Recently, using the generalization of Becker’s construction due to Betker [@Betker], Sugawa [@Sugawa2019] established a sufficient condition for a Beltrami coefficient in ${\mathbb{D}}$ to be 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 \[PR\_examples\] 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.

Relation between classes ${\mathcal{S}}_k$ and ${\mathcal{S}}_k^B$ {#SS_relation}
==================================================================

Although ${\mathcal{S}}_k^B$ represents only a part of ${\mathcal{S}}_k$, see e.g. [@Istvan §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. [@Kuhnau69; @Lehto71]. It is not difficult to see that $f_n\in{\mathcal{S}}_k^B$ for all $n\in{\mathbb{N}}$. Moreover, according to Proposition \[PR\_examples\], 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 $k_*\ge1/6$. Indeed, on the one hand, $|f''(z)/f'(z)|\le 6(1-|z|^2)$ for all $z\in{\mathbb{D}}$ and any $f\in{\mathcal{S}}$, see e.g. [@Goluzin 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. [@Lawrynowicz §22]. On the other hand, if a holomorphic function $f:{\mathbb{D}}\to{\mathbb{C}}$ satisfies $|f''(z)/f'(z)|\le k(1-|z|^2)$ for all $z\in{\mathbb{D}}$, then $f\in{\mathcal{S}}_k^B$, see [@Becker72 Satz 4.1].

We are able to improve slightly the estimate $k_*\ge 1/6$, see Corollary \[CR\_AW-Becker0\]. In this connection, it is natural to put forward the following problem.

\[PRO\_strong\] 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.

\[PRO\_weak\] 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 ${\mathrlap{\hskip.125em\widetilde{\hphantom{{\mathcal{S}}}\vphantom{\vbox to1.5ex{\vss}}}}{\mathcal{S}}_{k}^B\vphantom{\vbox to2.3ex{\vss}}}$ in the above problems as the following proposition shows.

For any $k\in(0,1)$, ${\mathrlap{\hskip.125em\widetilde{\hphantom{{\mathcal{S}}}\vphantom{\vbox to1.5ex{\vss}}}}{\mathcal{S}}_{k}^B\vphantom{\vbox to2.3ex{\vss}}}\subset{\mathcal{S}}^B_{\kappa(k)}$, where $\kappa(k):=2k/(1+k^2)$.

If a function $f\in{\mathrlap{\hskip.125em\widetilde{\hphantom{{\mathcal{S}}}\vphantom{\vbox to1.5ex{\vss}}}}{\mathcal{S}}_{k}^B\vphantom{\vbox to2.3ex{\vss}}}$ is generated by a Herglotz function $p$ satisfying , then the identity $$p_0\big(e^{i\Im Q(t)}z,\,\Re Q(t)\big)=L_t(p(z,t)),{~}{\text{~where~}}{~}L_t(z):=\frac{p(z,t)-i\Im p(0,t)}{\Re p(0,t)},$$ 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.$t\ge0$ and, moreover, generates the same function $f$. The latter can be verified using the change of variables $\tau:=Q(t)$, $\omega(\tau):=e^{i\Im Q(t)}w(t)$ that transforms the Loewner–Kufarev ODE  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 $0$. Taking into account that by , $H\big(p({\mathbb{D}},t)\big)\subset k\overline{\mathbb{D}}$ for a.e.$t\ge0$, 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.$t\ge0$ 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 \[PRO\_strong\] and \[PRO\_weak\] 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 $t\ge0$, and try to find out whether the map $F:{\mathbb{C}}\to{\mathbb{C}}$ defined by  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 ${\lim_{t\to+\infty}\rho(t)=+\infty}$.

1ex [**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 ${t\ge0}$, denote by $f^\Phi_t$, the conformal map of ${\mathbb{D}}$ onto $\Phi\big(\rho(t){\mathbb{D}}\big)$ normalized by ${f^\Phi_t(0)=0}$, ${\omega(t)(f^\Phi_t)'(0)>0}$. For a suitable choice of the function $\rho$, the family $(f^\Phi_t)_{t\ge0}$ is a Loewner chain. Using Courant’s Theorem, see e.g. [@Tsuji Theorem IX.14], it is possible to show that formula  defines a homeomorphism $F$ of $\C$. 1ex [**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 ${t\ge0}$, consider the conformal map $f_t^g$ of ${\mathbb{D}}$, ${f^g_t(0)=0}$, ${\omega(t)\big(f^g_t\big)'(0)>0}$, onto the Jordan domain bounded by ${g\big(\{z:|z|=\rho(t)\}\big)}$. For a suitable choice of the function $\rho$, the family $(f^g_t)_{t\ge0}$ is a Loewner chain and the map $F$ that it generates via  is a homeomorphism of $\C$. 1ex 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 $\omega$.

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 ${\omega:[0,+\infty)\to\partial{\mathbb{D}}}$.

Consider the function $$\label{EQ_f-example}
f(z):=\frac{2z(iz+\sqrt{1-z^2})^i}{1+\sqrt{1-z^2}}=\frac{2ze^{-\arcsin z}}{1+\sqrt{1-z^2}},\quad z\in{\mathbb{D}},$$ choosing the unique single-valued branch in ${\mathbb{D}}$ that belongs to ${\mathcal{S}}$. It is not difficult to check that $$\label{EQ_p}
p(z):=\frac{f(z)}{zf'(z)}=\sqrt{\frac{1+z}{1-z}}\quad\text{for all $z\in{\mathbb{D}}$}.$$ In particular, $\big|\arg p(z)\big|\le\pi/4$. Therefore, by a result of Betker [@Betker p.110], see also [@Ikkei-LM §5.1], $f$ can be extended to a $(1/\sqrt2)$-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 $\{tf(z):t\ge0\}$ 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. [@Ikkei-LM eq.(5.6) with $\lambda:=0$], is given by $$\Phi(z):=\frac{~r(1/\bar z)^2}{\hphantom{m}\vphantom{\int_0^1}\overline{f(1/\bar z)}\hphantom{m}}=\frac{4e^{-\pi}}{f(1/z)}\left(\frac{f(1/\bar z)}{|f(1/\bar z)|}\right)^{\! -2i\,\eta(z)}\!\!=~\frac{4e^{-\pi}}{f(1/z)}\left(\frac{f(1/z)}{f(1/\bar z)}\right)^{\! i\,\eta(z)}$$ for all $z\in\C\setminus{\mathbb{D}}$, where $\eta(z):={\mathop{\mathrm{sgn}}}\Im z$. Simple calculations give $$\frac{\Phi'_z(z)}{\Phi(z)}=\frac{1-i\eta(z)}{z^2}\frac{f'(1/z)}{f(1/z)},\quad \frac{\Phi'_{\bar z}(z)}{\Phi(z)}=\frac{i\eta(z)}{\bar{z}^2}\frac{f'(1/\bar z)}{f(1/\bar z)},\qquad |z|>1,~z\not\in{\mathbb{R}}.$$ 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 [@Betker_Hardy], there is no Loewner chain with image domains $D_r$ that defines a q.c.-extension via . Therefore, the homeomorphism $F$ in Construction 1 generated by the Loewner chain $(f^\Phi_t)$ is not quasiconformal, whichever $\rho$ and $\theta$ we choose.

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 $\{1/z\colon z\in{\overline{\mathbb{C}}}\setminus\overline{\Omega}\}=-\tfrac14\Omega.$ 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 [@Istvan proof of Theorem 2], we see that for all ${t\ge0}$ aside from some null-set $N$, ${\vphantom{f_t}\smash{\stackengine{-.075ex}{f_t}{\boldsymbol{\cdot\hskip.175em}}{O}{r}{F}{\useanchorwidth}{S}}}:=\partial f_t/\partial t$ and $f_t'$ exist a.e. on $\partial{\mathbb{D}}$, do not vanish, and $$\label{EQ_limit-property}
\frac{{\vphantom{f_t}\smash{\stackengine{-.075ex}{f_t}{\boldsymbol{\cdot\hskip.175em}}{O}{r}{F}{\useanchorwidth}{S}}}(e^{i\theta})}{e^{i\theta}f_t'(e^{i\theta})}\,\in\,U(k),$$ where $U(k)$ is defined in Theorem \[TH\_Becker\]. Moreover, by construction, $\partial f_t({\mathbb{D}})$ is $C^{\infty}$ when ${t>0}$. Hence, in fact, $f_t'$ extends smoothly to $\partial{\mathbb{D}}$ for all $t>0$; see e.g. [@Pommerenke:BB Chapter 3].

Taking into account that $\big|g^{-1}\big(f_t(e^{i\theta})\big)\big|=\rho(t)$ for all ${t\ge0}$ and all ${\theta\in[0,2\pi]}$, it follows that $\rho'(t)$ exists for any $t\in(0,+\infty)\setminus N$ and for the normal velocity of $\partial f_t({\mathbb{D}})$ we have $$|f'_t(e^{i\theta})|\,\Re\!\Big(\frac{{\vphantom{f_t}\smash{\stackengine{-.075ex}{f_t}{\boldsymbol{\cdot\hskip.175em}}{O}{r}{F}{\useanchorwidth}{S}}}(e^{i\theta})}{e^{i\theta}f'(e^{i\theta})}\Big)~=~\frac{\Re\big({\vphantom{f_t}\smash{\stackengine{-.075ex}{f_t}{\boldsymbol{\cdot\hskip.175em}}{O}{r}{F}{\useanchorwidth}{S}}}(e^{i\theta})\,\overline{e^{i\theta}f'(e^{i\theta})}\,\big)}{|f'_t(e^{i\theta})|}~=~\rho'(t)\,\big|g'\big(g^{-1}(f_t(e^{i\theta}))\big)\big|.$$ Together with  this implies that on the one hand, for any $t\in(0,+\infty)\setminus N$, $$\label{EQ_ravn}
\frac1K\le\left|\frac{\rho'(t)\,g'\big(g^{-1}(f_t(e^{i\theta}))}{f_t'(e^{i\theta})}\right|\le K:=\frac{1+k}{1-k}\quad \text{for all~$\theta\in[0,2\pi]$}.$$

On the other hand, $$\label{EQ_integration}
2\pi\rho(t)~=~\int_{0}^{2\pi}\left|\frac{{\mathrm{d}}g^{-1}(f_t(e^{i\theta}))}{{\mathrm{d}}\theta}\right|\,{\mathrm{d}}\theta~=~\int_{0}^{2\pi}\left|\frac{f_t'(e^{i\theta})}{g'\big(g^{-1}(f_t(e^{i\theta}))}\right|\,{\mathrm{d}}\theta$$

Combining  with , we see that $$\frac{\rho(t)}{K^2}\le \left|\frac{f_t'(e^{i\theta})}{g'\big(g^{-1}(f_t(e^{i\theta}))}\right|\le \rho(t) K^2\quad t>0,~t\not\in N.$$ 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. [@Pommerenke:BB p.18]) and Courant’s Theorem (see e.g. [@Tsuji Theorem IX.14]) we conclude that $\gamma_t\to\gamma_0$ as $t\to0^+$. 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.

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 . This simple result is probably known to specialists: somewhat similar ideas appeared e.g. in [@Betker_Phi] and [@Ikkei1 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 [@Shabat §15, §8].

\[TH\_PDE\_THM\] Let $f$ be a holomorphic function in $\D$, with $f'(0)-1=f(0)=0$. Suppose that there exists a meromorphic solution $\Phi : \C \times \D \to \C$ to the PDE initial value problem $$\begin{aligned}
 &\Phi'_w(z, \,w ) = \varphi(z,w)\, \Phi'_z(z, \,w), & (z,w)\in \C \times \D;\label{EQ_PDE1}\\
 &\Phi(z,z)  = f(z), & z\in{\mathbb{D}},\label{EQ_PDE2}\end{aligned}$$ with a coefficient $\varphi$ meromorphic in $\C \times \D$ and satisfying the following two conditions:

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

-   $r\!\left|\varphi(w/r,w)\right| \le k$ for all $w\in\D$ and all $r\in\big(|w|^2,1\big)$.

Suppose also that there exists $\varepsilon\in(0,1)$ and $M>0$ such that $$\label{EQ_Phi-at-infinity}
|\Phi(z,w)|\le M |z|\quad\text{whenever~$|w|\le |z|$ and $|z\,w|\le \varepsilon^2$.}$$ Then $f$ admits a $k$-q.c. Becker extension given by $$F(z) := \Phi(z, 1/\bar{z}), \hspace{15pt} |z|>1.$$ In particular, $f\in{\mathcal{S}}_k^B$.

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

Before proving Theorem \[TH\_PDE\_THM\], let us consider a few examples.

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

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

The following corollary represents another example.

\[CR\_AW-Becker\] Fix $k\in(0,1)$. Let $f(z)=z+a_2 z^2+\ldots$ be holomorphic in ${\mathbb{D}}$. If $$\label{EQ_AW-Becker}
\tfrac{4\sqrt3}{9}(1-|z|^2)|a_2|\,+\,(1-|z|^2)^2\big|a_2^2+\tfrac12S_f(z)\big|~\le~ k\quad\text{for all~$z\in{\mathbb{D}}$},$$ 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 $$\label{EQ_Phi}
\Phi(z,w):=f(w)+\frac{f'(w)}{\frac{1}{z-w}+a_2-\frac{1}{2}\frac{f''(w)}{f'(w)}}.$$

Let $\varphi(z,w):=2a_2(z-w)+(z-w)^2\big(a_2^2+\tfrac12S_f(w)\big)$, where $S_f$ stands for the Schwarzian derivative of $f$. Then $\Phi$ given by  solves problem ,.

Moreover, there exists $K>1$ such that $\big|f'(w)\big|\le K$ and $\left|a_2-\frac12\big({f''(w)}/{f'(w)}\big)\right|\le K|w|$ for all $w\in\frac12{\mathbb{D}}$. Hence, for any $(z,w)\in\C\times{\mathbb{D}}$ with $|w|\le |z|$ and ${|zw|\le\varepsilon^2}:=(4K)^{-1}$, $$\big|\Phi(z,w)\big|\,\le\,\big|f(w)\big|\,+\,\frac{\big|(z-w)f'(w)\big|}{\left|1 - K|w|\cdot|z-w|\vphantom{\int_0^1}\right|} \,\le\, K|w|\,+\,\frac{2\,|z|\,\big|f'(w)\big|}{1 - 2K\varepsilon^2\,} \,\le\, 5 K|z|.$$ This proves . Finally, since $|w|(1-|w|^2)\le 2\sqrt 3/9$ for all $w\in{\mathbb{D}}$, condition  ensures that $\varphi$ satisfies (ii), while (i) holds trivially.

Thus, the desired conclusion takes place due to Theorem \[TH\_PDE\_THM\].

A well-known result by Ahlfors and Weill [@AW62], see also [@Ahlfors74], asserts that if a holomorphic function $f:{\mathbb{D}}\to{\mathbb{C}}$ satisfies $\frac12(1-|z|^2)^2|S_f(z)|\le k$, 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., [@Becker72 §4] and [@Becker80]), but it does not have to fix $\infty$ and hence $F$ is not a Becker extension in general (which was overlooked in [@Istvan §5]). Corollary \[CR\_AW-Becker\] 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 \[CR\_AW-Becker\] coincides with the extension constructed by Ahlfors and Weill [@Ahlfors74].

It is known, see e.g. [@Lawrynowicz Example 4 on p.132], that given $k\in(0,1)$, for all $f\in{\mathcal{S}}_k$, $|a_2|\le 2k$ and $|S_f(z)|\le 6k/(1-|z|^2)^2$ for any $z\in{\mathbb{D}}$. Therefore, Corollary \[CR\_AW-Becker\] implies immediately the following statement.

\[CR\_AW-Becker0\] If $0<k<0.188856\ldots$, then ${\mathcal{S}}_k\subset {\mathcal{S}}_q^B$ with $q:=(3+\tfrac{8\sqrt3}{9})k+4k^2$.

It remains to prove the main result of this section.

Let $$f_t(\zeta):=\Phi(\zeta e^t,\zeta e^{-t})\quad \text{and}\quad p(\zeta,t):=\frac{\partial f_t(\zeta)/\partial t}{\zeta f_t'(\zeta)}\quad\text{for all $t\ge0$ and~$\zeta\in{\mathbb{D}}$.}$$ Thanks to condition , these functions are holomorphic in $\zeta\in\varepsilon{\mathbb{D}}$ and real-analytic in $t\ge0$, and moreover, $|f_t(\zeta)|\le M e^t$ for all $\zeta\in\varepsilon{\mathbb{D}}$ and $t\ge0$.

Note also that for any fixed $t\ge0$, 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 $\C\times{\mathbb{D}}$, we would have that $(\zeta e^t,\zeta e^{-t})\in\mathcal P$ for all $\zeta\in{\mathbb{D}}$, which contradicts . Therefore, $p(\cdot,t)$ and $f_t$ are well-defined meromorphic functions in ${\mathbb{D}}$ for each $t\ge0$.

As an elementary calculation shows, for all $\zeta\in{\mathbb{D}}$ and $t>0$, $$\frac{1-p(\zeta,t)}{1+p(\zeta,t)}=e^{-2t}\,\frac{\Phi'_w(\zeta e^t,\zeta e^{-t})}{\Phi'_z(\zeta e^t,\zeta e^{-t})}
=
r\,\varphi(w/r,w),$$ 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 .

Clearly, $f_t(0)=\Phi(0,0)=f(0)=0$, $t\ge0$. Moreover, taking into account (i), we have $f'_t(0)=e^t\Phi'_z(0,0)+e^{-t}\Phi'_w(0,0)=e^t\Phi'_z(0,0)=e^t f'(0)=e^t$ for all $t\ge0$.

We see that $(f_t)$ satisfies the hypothesis of Pommerenke’s Criterion [@Pommerenke Theorem 6.1 on p.159]. Hence $(f_t)$ is a classical radial Loewner chain. Furthermore, by Theorem \[TH\_Becker\], $f=f_0$ admits a $k$-q.c. extension $F:{\mathbb{C}}\to{\mathbb{C}}$ given by the formula $$F(e^{t+i\theta})=f_t(e^{i\theta})=\Phi(e^{t}e^{i\theta},e^{-t}e^{i\theta})=\Phi(z,1/\bar z)\quad \text{for all $z:=e^{t+i\theta}\in{\mathbb{C}}\setminus\overline{\mathbb{D}}$,}$$ which was to be proved.

Acknowledgement {#acknowledgement .unnumbered}
===============

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 [@Sugawa2019].

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