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}
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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 