ar X iv : m at h / 04 10 57 8 v 1 [ m at h . C V ] 2 7 O ct 2 00 4 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 + a2z + . . . in U , and let S be its subclass defined by restricting |f(z)| < M in U , M ≥ 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 . Using Löwner’s parametric representation and the optimal control method we give exact initial Bombieri’s numbers and derive a sharp constant M0, such that for all M ≥ M0 the Pick function gives the local maximum to |a4|. Numerical approximation is given. §