Local extremal problems for bounded analytic functions without zeros

D. V. Prokhorov ; S. V. Romanova

Izvestiya. Mathematics , Tome 70 (2006) no. 4, pp. 841-856

Résumé

In the class $B(t)$, $t>0$, of all functions $f(z,t)=e^{-t}+c_1(t)z+c_2(t)z^2+\dots$ that are analytic in the unit disc $U$ and such that $0|f(z,t)|1$ in $U$, we obtain asymptotic estimates for the coefficients for small and sufficiently large $t>0$. We suggest an algorithm for determining those $t>0$ for which the canonical functions provide the local maximum of $\operatorname{Re}c_n(t)$ in $B(t)$. We describe the set of functionals $L(f)=\sum_{k=0}^n\lambda_kc_k$ for which the canonical functions provide the maximum of $\operatorname{Re}L(f)$ in $B(t)$ for small and large values of $t$. The proofs are based on optimization methods for solutions of control systems of differential equations.