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}
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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 