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 