extensions: Becker’s class and estimates of the third coefficient.'
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Introduction
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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 