Herglotz function $p$ satisfies $$\label{EQ_Becker-condition}
  p({\mathbb{D}},t) \subset U(k):=\left\{ w \in \C \colon \left|\frac{w-1}{w+1}\right| \le k\right\}\quad\text{for  a.e.~$t\ge0$}.$$ Then for every $t\ge0$, the function $f_t$ admits a $k$-q.c. extension to ${\overline{\mathbb{C}}}$ that fixes $\infty$. In particular, such an extension for $f_0$ is given by $$\label{EQ_BeckerExt}
F(\rho e^{i\theta}):=\left\{
 \begin{array}{ll}
  f_0(\rho e^{i\theta}), & \text{if~$0\le \rho<1$},\\[1ex]
  f_{\log \rho}(e^{i\theta}), & \text{if~$\rho\ge1$}.
 \end{array}
\right.$$

\[RM\_2018\] According to [@Istvan Theorem 2], a sort of converse statement holds. Namely, if $(f_t)$ is a Loewner chain such that all $f_t$’s extend continuously to $\partial{\mathbb{D}}$ and the map $F$ defined by  is $k$-quasiconformal in $\C$, then the Herglotz function $p$ associated with $(f_t)$ satisfies Becker’s condition .

In what follows, for $k\in(0,1)$, we will denote by ${\mathcal{S}}_k^B$ the class of all $f\in{\mathcal{S}}$ admitting Loewner’s representation with the Herglotz function $p$ normalized by $p(0,t)=1$ a.e.$t\ge0$ and satisfying . A bit larger class of all $f\in{\mathcal{S}}$ generated by Herglotz functions subject to Becker’s condition , but not necessarily normalized, will be denoted by ${\mathrlap{\hskip.125em\widetilde{\hphantom{{\mathcal{S}}}\vphantom{\vbox to1.5ex{\vss}}}}{\mathcal{S}}_{k}^B\vphantom{\vbox to2.3ex{\vss}}}$.

According to Theorem \[TH\_Becker\], ${\mathcal{S}}^B_k\subset{\mathrlap{\hskip.125em\widetilde{\hphantom{{\mathcal{S}}}\vphantom{\vbox to1.5ex{\vss}}}}{\mathcal{S}}_{k}^B\vphantom{\vbox to2.3ex{\vss}}}\subset{\mathcal{S}}_k$. It is known that ${\mathrlap{\hskip.125em\widetilde{\hphantom{{\mathcal{S}}}\vphantom{\vbox to1.5ex{\vss}}}}{\mathcal{S}}_{k}^B\vphantom{\vbox to2.3ex{\vss}}}\neq{\mathcal{S}}_k$, see . However, it seems that the study of ${\mathcal{S}}_k^B$ and ${\mathcal{S}}_k$ is still of considerable interest. It is worth to mention that 