is known, see e.g. [@Lawrynowicz Example 4 on p.132], that given $k\in(0,1)$, for all $f\in{\mathcal{S}}_k$, $|a_2|\le 2k$ and $|S_f(z)|\le 6k/(1-|z|^2)^2$ for any $z\in{\mathbb{D}}$. Therefore, Corollary \[CR\_AW-Becker\] implies immediately the following statement.

\[CR\_AW-Becker0\] If $0<k<0.188856\ldots$, then ${\mathcal{S}}_k\subset {\mathcal{S}}_q^B$ with $q:=(3+\tfrac{8\sqrt3}{9})k+4k^2$.

It remains to prove the main result of this section.

Let $$f_t(\zeta):=\Phi(\zeta e^t,\zeta e^{-t})\quad \text{and}\quad p(\zeta,t):=\frac{\partial f_t(\zeta)/\partial t}{\zeta f_t'(\zeta)}\quad\text{for all $t\ge0$ and~$\zeta\in{\mathbb{D}}$.}$$ Thanks to condition , these functions are holomorphic in $\zeta\in\varepsilon{\mathbb{D}}$ and real-analytic in $t\ge0$, and moreover, $|f_t(\zeta)|\le M e^t$ for all $\zeta\in\varepsilon{\mathbb{D}}$ and $t\ge0$.

Note also that for any fixed $t\ge0$, the point $(\zeta e^t,\zeta e^{-t})$ can lie in the polar set $\mathcal P$ of $\Phi$ only for $\zeta$ belonging to a discrete subset of ${\mathbb{D}}$. Otherwise, since $\mathcal P$ is an analytic set in $\C\times{\mathbb{D}}$, we would have that $(\zeta e^t,\zeta e^{-t})\in\mathcal P$ for all $\zeta\in{\mathbb{D}}$, which contradicts . Therefore, $p(\cdot,t)$ and $f_t$ are well-defined meromorphic functions in ${\mathbb{D}}$ for each $t\ge0$.

As an elementary calculation shows, for all $\zeta\in{\mathbb{D}}$ and $t>0$, $$\frac{1-p(\zeta,t)}{1+p(\zeta,t)}=e^{-2t}\,\frac{\Phi'_w(\zeta e^t,\zeta e^{-t})}{\Phi'_z(\zeta e^t,\zeta e^{-t})}
=
r\,\varphi(w/r,w),$$ where $r:=e^{-2t}$ and $w:=\zeta e^{-t}$. Trivially, $|w|^2<r<1$. Therefore, condition (ii) implies that $p$ is a Herglotz function satisfying Becker’s condition .

Clearly, $f_t(0)=\Phi(0,0)=f(0)=0$, $t\ge0$. Moreover, taking into account (i), 