form $k\overline{\phi(z)}/|\phi(z)|$, where $\phi$ is holomorphic, see [@HarringtonOrtel p.161]. On the other hand, in terms of the Laurent development $\varphi_n(z)=\sum_{m=3}^{+\infty}c_{n,m}z^{-m}$, we have $$\frac{\Lambda(\varphi_n)}{2\pi}=\frac{1+\log\rho(k)}{\rho(k)}c_{n,3}~+~\frac{\rho(k)^2-2\log\rho(k)-1}{2}~\sum_{m=4}^{+\infty}
\frac{(-1)^m c_{n,m}}{\rho(k)^{m-2}}~\text{~}\longrightarrow~0~\text{~}\text{as~}~n\to+\infty,$$ because for a fixed $r\in\big(1,\rho(k)\big)$ the Cauchy estimates give $|c_{n,m}|\le r^{m}\max_{|z|=r}|\varphi_n(z)|$. We obtained a contradiction, which shows that $f_+$ has a $q$-q.c. extension to $\C$ with ${q\in(0,k)}$, and hence the proof is complete.

Extremal Becker extensions {#SS_examples}
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Recall that a q.c.-extension $F:\C\to\C$ of a function $f\in{\mathcal{S}}$ is called *extremal*, if for any q.c.-extension $G:{\mathbb{C}}\to{\mathbb{C}}$ of $f$ we have ${\mathop{\mathrm{ess\,sup}}}_{|z|>1}|\mu_G(z)|\ge{\mathop{\mathrm{ess\,sup}}}_{|z|>1}|\mu_F(z)|$, where $\mu_G$ and $\mu_F$ stand for the Beltrami coefficients of $G$ and $F$, respectively. If the equality occurs in the above inequality only for $G=F$, then $F$ is said to be the *uniquely extremal* q.c.-extension of $f$ to ${\mathbb{C}}$.

There is a simple sufficient condition for a q.c.-extension $F:\C\to\C$ to be uniquely extremal. A *(regular) Teichmüller mapping* of a domain $D$ is a q.c-mapping $F:D\to{\overline{\mathbb{C}}}$ such that $\mu_F(z)=k\overline{\varphi(z)}/{|\varphi(z)|}$ for a.e. $z\in D$, where $k\in(0,1)$ and ${\varphi(z)\,{\mathrm{d}}z^2}$, ${\varphi\not\equiv0}$, is a holomorphic quadratic differential in $D$. It is known that [@Strebel:1978 Theorem 4] if a q.c.-extension of $f\in{\mathcal{S}}$ to $\C$ is Teichmüller on $\Delta:=\C\setminus\overline{\mathbb{D}}$ with $\varphi$ satisfying $\|\varphi\|:={\iint_\Delta|\varphi(z)|\,{\mathrm{d}}x{\mathrm{d}}y<+\infty}$, then $F$ is uniquely extremal.

If $\varphi$ is holomorphic 