**Dmitri Prokhorov, Alexander Vasil'ev**

## Optimal Control in
Bombieri's and Tammi's Conjectures

**Abstract:**

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.

**Keywords:**

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

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