A Problem of Bombieri on Univalent Functions

TL;DR

This paper revisits Bombieri's problem on the coefficients of univalent functions, exploring its historical context and proposing a new approach inspired by de Branges' methods, with potential implications for complex analysis.

Contribution

It offers a historical overview of Bombieri's problem and introduces a revised version and a novel approach inspired by de Branges' techniques.

Findings

Connected Bombieri's problem to de Branges' methods

Proposed a revised formulation of Bombieri's problem

Suggested new directions for research in univalent functions

Abstract

The famous Bieberbach Conjecture from 1916 on the coefficients of normalized univalent functions defined in the unit disk that was finally proved by de Branges some 70 years later, drifted many complex analysts attention to other subjects. Those who continued to explore de Branges method and push it as far as possible were not aware of where it may lead. Surprisingly enough, a paper by X. H. Dong that fell in our hands contained a way to tackle one of the problems of Bombieri on the behavior of the coefficients of univalent functions. We shall give an account of the history of the problem and a revised version of it.