is clear that $f_{-}(z)=-f_+(-z)$, and the set of all extremal functions for $|a_3|$ coincides with the rotations of $f_+$. Therefore, we may assume the sign “$+$” in . Then the same method as in case $a=0$ allows us to write down the corresponding Heglotz function explicitly, $$p(z,t)=\frac{1-k z^2+(1-k)e^{t-t_0}z}{1+kz^2+(1+k)e^{t-t_0}z}~\text{~for $t\in[0,t_0]$}\quad
\text{and}\quad p(z,t)=\frac{1 - kz}{1 + kz}~\text{~for~$t\ge t_0$}.$$

Unfortunately, it does not seem possible to get an explicit formula for the extremal function $f_+$ and the Loewner chain generated by the above Herglotz function. However, one can find the Beltrami coefficient of the Becker extension provided by this Loewner chain, see e.g. [@Istvan Proof of Theorem 2], $$\label{EQ_extremal-Beltrami-coefficient}
\mu(z)=\frac{z^2}{|z|^2}\,\frac{p\big(z/|z|,\log |z|\big)-1}{p\big(z/|z|,\log |z|\big)+1}=
\begin{cases}
-k\dfrac{z^4}{|z|^4}\,\dfrac{\rho(k) + \bar z}{\rho(k) + z},&\text{if $|z|\in\big(1,\,\rho(k)\big)$,}\\[2.75ex]
-k\dfrac{z^3}{|z|^3}, & \text{if $|z|>\rho(k)$,}
\end{cases}$$ where $\rho(k):=e^{t_0}=\exp\big((1/k-1)/2\big)$.

Note that $|a_3|\le|a_3-\alpha a_2^2|+\alpha |a_2|^2$ for any $\alpha\in(0,1)$. The inequality $\max_{{\mathcal{S}}_k}|a_3|\le\varrho(k)$ follows therefore from the Fekete–Szegő Theorem, see e.g. [@Duren p.104], the well-known estimate $|a_2|\le 2$ for the class ${\mathcal{S}}$, and Lehto’s Majorant Principle [@Lehto_Majorant].

To show that the maximum of $|a_3|$ in ${\mathcal{S}}_k$ is strictly greater than in ${\mathcal{S}}_k^B$, fix $k\in(0,1)$ and note that for any non-constant holomorphic functional $\Phi:{\mathcal{S}}\to{\mathbb{C}}$, according to Lehto’s Majorant Principle, the function $q\mapsto\max_{{\mathcal{S}}_q}|\Phi|$ is *strictly* increasing. It follows that the 