\[PR\_examples\] 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}
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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 