a_2(0)=0,\label{EQ_phase1}\\[.7ex]
\frac{{\mathrm{d}}a_3}{{\mathrm{d}}t}&=-e^{-2t}p_2(t)-2e^{-t}p_1(t)a_2(t)\notag\\&=-k\Big(e^{-2t}\big(c_2(t)-(1-k)\frac{c_1(t)^2}{2}\,\big)+2e^{-t}c_1(t)a_2(t)\Big), & a_3(0)=0.\label{EQ_phase2}\end{aligned}$$

Since along with any $f\in{\mathcal{S}}_k^B$ the class ${\mathcal{S}}_k^B$ contains all rotations of $f$, i.e. the functions ${z\mapsto e^{i\theta}f(e^{-i\theta}z)}$, ${\theta\in{\mathbb{R}}}$, the problem to determine $\max|a_3|$ in ${\mathcal{S}}_k^B$ is equivalent to finding $~\max\,\Re a_3$. The latter problem can be reformulated as the optimal control problem for the above system and the objective functional $\Re a_3(+\infty)$, with a control function ${t\mapsto \big(c_1(t),c_2(t)\big)\in\C^2}$ regarded as admissible if it is measurable and for a.e. $t\ge0$ satisfies $$\label{EQ_Cara}
 |c_1|\le 2,\qquad |2 c_2-c_1^2|\le 4-|c_1|^2.$$ Conditions  describe the value region of ${\mathcal{C}}\ni q\mapsto (c_1,c_2)\in\C^2$ over the Carathéodory class ${\mathcal{C}}$ of all holomorphic functions $q(z)=1+c_1 z+c_2 z^2+\ldots$ in ${\mathbb{D}}$ with positive real part; see, e.g., [@Tsuji Chapter IV, §7].

To apply Pontryagin’s Maximum Principle, we define the (holomorphic) Hamiltonian $$H(a_2,a_3,\psi_2,\psi_3,t,c_1,c_2):=-ke^{-t} c_1\psi_2-k\Big(e^{-2t}\big(c_2-(1-k)\frac{c_1^2}{2}\big)+2e^{-t}c_1a_2\Big)\psi_3$$ and write the adjoint system $$\begin{aligned}
\frac{{\mathrm{d}}\psi_2}{{\mathrm{d}}t}&=-\frac{\partial H}{\partial a_2}=2ke^{-t}c_1(t)\psi_3(t),\label{EQ_adj1}\\[1.25ex]
\frac{{\mathrm{d}}\psi_3}{{\mathrm{d}}t}&=-\frac{\partial H}{\partial a_3}=0.\label{EQ_adj2}\end{aligned}$$ The maximum of $\Re a_3(+\infty)$ is to be found among all the trajectories of , satisfying the initial condition at $t=0$, while the right-hand endpoint of the trajectories is variable. Therefore, according to Pontryagin’s Maximum Principle, see [@Pontryagin Chapter I, §7, Theorem $3^*$], if $c_1(t)=c_1^*(t)$, $c_2(t)=c_2^*(t)$ is an optimal control in our problem, then for the corresponding solution to the phase system 