The Bombieri Problem for Bounded Univalent Functions

Abstract

Bombieri proposed to describe the structure of the sets of values of the initial coefficients of normalized conformal mappings of the disk in a neighborhood of the corner point corresponding to the Koebe function. The Bombieri numbers characterize the limit position of the support hyperplane passing through a critical corner point. In this paper, the Bombieri problem is studied for the class of bounded normalized conformal mappings of the disk, where the role of the Koebe function is played by the Pick function. The Bombieri numbers for a pair of two nontrivial initial coefficients are calculated.