in $\Delta$ and has a zero of order at least four at $\infty$, then a q.c.-map of $\Delta$ with the Beltrami coefficient $k\overline\varphi/|\varphi|$ is a Teichmüller mapping of the *simply connected* domain ${\overline{\mathbb{C}}}\setminus\overline{\mathbb{D}}$. For this case, certain conditions weaker than $\|\varphi\|<+\infty$ are sufficient for (unique) extremality, see e.g. [@Huang; @YaoF; @YaoP] and references therein.

Using the above mentioned sufficient condition, we construct a quite large family of functions $f\in{\mathcal{S}}_k$ with uniquely extremal extensions obtained via Becker’s construction. The idea comes from the following example. Consider the function ${f_\sigma\in{\mathcal{S}}}$, ${\sigma\in(0,2)}$, obtained by composing ${\mathbb{H}}:=\{z\colon \Re z>0\}\ni \zeta\mapsto\zeta^\sigma$, $1\mapsto 1$, with suitable Moebius transformations. This function admits a unique $|\sigma-1|$-q.c. extension $F_\sigma:\C\to\C$ and belongs to ${\mathrlap{\hskip.125em\widetilde{\hphantom{{\mathcal{S}}}\vphantom{\vbox to1.5ex{\vss}}}}{\mathcal{S}}_{|\sigma-1|}^B\vphantom{\vbox to2.3ex{\vss}}}$, see [@Istvan Example 2]. The Beltrami coefficient of $F_\sigma$ is $\mu(z)={(\sigma-1)\overline{\varphi(z)}/|\varphi(z)|}$, ${\varphi(z):=1/(z^2-1)^2}$, for all $z\in\Delta$, which can be written as $\mu(\rho\zeta)=\zeta^2\,\psi_\rho(\zeta)$ for all $\rho>1$ and $\zeta\in\partial{\mathbb{D}}$, where $\psi_\rho(\zeta):=(\sigma-1)(\zeta^2-1/\rho^2)/(1-\zeta^2/\rho^2)$. The latter means that $F$ is Becker’s q.c.-extension  with the Herglotz function $p(z,t):=\big(1-\psi_{e^t}(z)\big)/\big(1+\psi_{e^t}(z)\big)$. Note that, up to the factor $(\sigma-1)$, $\psi_\rho$ is a Blaschke product. It turns out that any finite Blaschke product gives rise to a similar example.

\[PR\_examples\] Let $k\in(0,1)$, $n\in{\mathbb{N}}$, $a_1,\ldots, a_n\in{\mathbb{D}}$, $\alpha\in{\mathbb{R}}$. Then the Herglotz function $$\label{EQ_examples_p}
  p(z,t):=\frac{1+k\psi_t(z)}{1-k\psi_t(z)},~\text{~where~}~\psi_t(z):=e^{i\alpha}\, \prod_{j=1}^n\frac{z-e^{-t}a_j}{1-e^{-t}\,\overline a_j z},\quad z\in{\mathbb{D}},~t\ge0,$$ 