one hand, $|f''(z)/f'(z)|\le 6(1-|z|^2)$ for all $z\in{\mathbb{D}}$ and any $f\in{\mathcal{S}}$, see e.g. [@Goluzin Ch.II,§4, ineq.(6)], with $6$ replaced by $6k$ if $f\in{\mathcal{S}}_k$ thanks to Lehto’s Majorant Principle, see e.g. [@Lawrynowicz §22]. On the other hand, if a holomorphic function $f:{\mathbb{D}}\to{\mathbb{C}}$ satisfies $|f''(z)/f'(z)|\le k(1-|z|^2)$ for all $z\in{\mathbb{D}}$, then $f\in{\mathcal{S}}_k^B$, see [@Becker72 Satz 4.1].

We are able to improve slightly the estimate $k_*\ge 1/6$, see Corollary \[CR\_AW-Becker0\]. In this connection, it is natural to put forward the following problem.

\[PRO\_strong\] Find $k_*$. In particular, is it true that $k_*=1$, i.e. that for any $k\in(0,1)$ there exists $q\in(0,1)$ such that ${\mathcal{S}}_k\subset{\mathcal{S}}_q^B$?

It seems interesting to consider also a bit weaker version of the latter question.

\[PRO\_weak\] Is it true that for any function $f\in{\mathcal{S}}$ admitting a q.c.-extension to ${\mathbb{C}}$, there exists $q\in(0,1)$, possibly depending on $f$, such that $f\in{\mathcal{S}}_q^B$?

Note that it is possible to replace ${\mathcal{S}}_k^B$ with ${\mathrlap{\hskip.125em\widetilde{\hphantom{{\mathcal{S}}}\vphantom{\vbox to1.5ex{\vss}}}}{\mathcal{S}}_{k}^B\vphantom{\vbox to2.3ex{\vss}}}$ in the above problems as the following proposition shows.

For any $k\in(0,1)$, ${\mathrlap{\hskip.125em\widetilde{\hphantom{{\mathcal{S}}}\vphantom{\vbox to1.5ex{\vss}}}}{\mathcal{S}}_{k}^B\vphantom{\vbox to2.3ex{\vss}}}\subset{\mathcal{S}}^B_{\kappa(k)}$, where $\kappa(k):=2k/(1+k^2)$.

If a function $f\in{\mathrlap{\hskip.125em\widetilde{\hphantom{{\mathcal{S}}}\vphantom{\vbox to1.5ex{\vss}}}}{\mathcal{S}}_{k}^B\vphantom{\vbox to2.3ex{\vss}}}$ is generated by a Herglotz function $p$ satisfying , then the identity $$p_0\big(e^{i\Im Q(t)}z,\,\Re Q(t)\big)=L_t(p(z,t)),{~}{\text{~where~}}{~}L_t(z):=\frac{p(z,t)-i\Im p(0,t)}{\Re p(0,t)},$$ and $Q(t):=\int_0^t\!p(0,s){\mathrm{d}}s$, defines a Herglotz function $p_0$ that obeys the normalization $p_0(0,t)=1$ 