$$\label{EQ_integration}
2\pi\rho(t)~=~\int_{0}^{2\pi}\left|\frac{{\mathrm{d}}g^{-1}(f_t(e^{i\theta}))}{{\mathrm{d}}\theta}\right|\,{\mathrm{d}}\theta~=~\int_{0}^{2\pi}\left|\frac{f_t'(e^{i\theta})}{g'\big(g^{-1}(f_t(e^{i\theta}))}\right|\,{\mathrm{d}}\theta$$

Combining  with , we see that $$\frac{\rho(t)}{K^2}\le \left|\frac{f_t'(e^{i\theta})}{g'\big(g^{-1}(f_t(e^{i\theta}))}\right|\le \rho(t) K^2\quad t>0,~t\not\in N.$$ Therefore, the conformal weldings $\gamma_t:=\big(g^{-1}\circ f_t|_{\partial{\mathbb{D}}}\big)\big/\rho(t)$, $t\in(0,+\infty)$, are $K^2$-Lipschitz continuous. Using Carathéodory’s Extension Theorem (see e.g. [@Pommerenke:BB p.18]) and Courant’s Theorem (see e.g. [@Tsuji Theorem IX.14]) we conclude that $\gamma_t\to\gamma_0$ as $t\to0^+$. It follows that $\gamma_0$ has to be also Lipschitz-continuous, but in reality it is not. This contradiction shows that $F$ is not quasiconformal.

A sufficient condition for Becker extendibility
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Below we prove a sufficient condition for a holomorphic function to be Becker extendible, i.e. to have a q.c.-extension of the form . This simple result is probably known to specialists: somewhat similar ideas appeared e.g. in [@Betker_Phi] and [@Ikkei1 equation(11)]. However, it does not seem to be ever stated in the form as presented below. For the notions of a meromorphic function of several complex variables and that of an analytic set we refer the reader to [@Shabat §15, §8].

\[TH\_PDE\_THM\] Let $f$ be a holomorphic function in $\D$, with $f'(0)-1=f(0)=0$. Suppose that there exists a meromorphic solution $\Phi : \C \times \D \to \C$ to the PDE initial value problem $$\begin{aligned}
 &\Phi'_w(z, \,w ) = \varphi(z,w)\, \Phi'_z(z, \,w), & (z,w)\in \C \times 