$f$. Then $\Phi$ given by  solves problem ,.

Moreover, there exists $K>1$ such that $\big|f'(w)\big|\le K$ and $\left|a_2-\frac12\big({f''(w)}/{f'(w)}\big)\right|\le K|w|$ for all $w\in\frac12{\mathbb{D}}$. Hence, for any $(z,w)\in\C\times{\mathbb{D}}$ with $|w|\le |z|$ and ${|zw|\le\varepsilon^2}:=(4K)^{-1}$, $$\big|\Phi(z,w)\big|\,\le\,\big|f(w)\big|\,+\,\frac{\big|(z-w)f'(w)\big|}{\left|1 - K|w|\cdot|z-w|\vphantom{\int_0^1}\right|} \,\le\, K|w|\,+\,\frac{2\,|z|\,\big|f'(w)\big|}{1 - 2K\varepsilon^2\,} \,\le\, 5 K|z|.$$ This proves . Finally, since $|w|(1-|w|^2)\le 2\sqrt 3/9$ for all $w\in{\mathbb{D}}$, condition  ensures that $\varphi$ satisfies (ii), while (i) holds trivially.

Thus, the desired conclusion takes place due to Theorem \[TH\_PDE\_THM\].

A well-known result by Ahlfors and Weill [@AW62], see also [@Ahlfors74], asserts that if a holomorphic function $f:{\mathbb{D}}\to{\mathbb{C}}$ satisfies $\frac12(1-|z|^2)^2|S_f(z)|\le k$, where $k\in(0,1)$, for all $z\in{\mathbb{D}}$, then $f$ is univalent and extends to a $k$-q.c. automorphism $F:{\overline{\mathbb{C}}}\to{\overline{\mathbb{C}}}$, with $F$ given by an explicit formula. This extension can be obtained with the help of Becker’s construction (see, e.g., [@Becker72 §4] and [@Becker80]), but it does not have to fix $\infty$ and hence $F$ is not a Becker extension in general (which was overlooked in [@Istvan §5]). Corollary \[CR\_AW-Becker\] is a sort of modification of the Ahfors–Weill condition that ensures extendibility to a q.c.-automorphism of ${\mathbb{C}}$. In fact, if $a_2=0$ then the q.c.-extension of $f$ given in Corollary \[CR\_AW-Becker\] coincides with the extension constructed by Ahlfors and Weill [@Ahlfors74].

It 