Dmitri Prokhorov, Alexander Vasil'ev

Optimal Control in Bombieri's and Tammi's Conjectures

Let $S$ stand for the usual class of univalent regular functions in the unit disk $U=\{z:|z|<1\}$ normalized by $f(z)=z+a_2z^2+\dots$ in $U$, and let $S^M$ be its subclass defined by restricting $|f(z)|<M$ in $U$, $M\geq 1$. We consider two classical problems: Bombieri's coefficient problem for the class $S$ and the sharp estimate of the fourth coefficient of a function from $S^M$. Using L\"owner's parametric representation and the optimal control method we propose a way to finding initial Bombieri's numbers and derive a sharp constant $M_0$ such that for all $M\geq M_0$ the Pick function gives the local maximum to $|a_4|$. Numerical approximation is given.

Extremal problem, bounded univalent function, Bombieri's conjecture, optimal control.

MSC 2000: Primary: 30C50. Secondary: 30C70, 49K15