Bieberbach conjecture, Bohr radius, Bloch constant and Alexander's theorem in infinite dimensions

TL;DR

This paper explores properties of holomorphic mappings in infinite-dimensional complex Banach spaces, establishing criteria for geometric features and extending classical theorems like Bieberbach and Alexander's to this setting.

Contribution

It introduces new criteria for univalence, starlikeness, and quasi-convexity, and generalizes key classical results to infinite-dimensional holomorphic mappings.

Findings

Criteria for univalence, starlikeness, quasi-convexity established

Generalized Bieberbach conjecture and covering theorem proved

Lower bounds for Bloch constant and Alexander's theorem extended

Abstract

In this paper, we investigate holomorphic mappings $F$ on the unit ball $\mathbb{B}$ of a complex Banach space of the form $F(x)=f(x)x$, where $f$ is a holomorphic function on $\mathbb{B}$. First, we investigate criteria for univalence, starlikeness and quasi-convexity of type $B$ on $\mathbb{B}$. Next, we investigate a generalized Bieberbach conjecture, a covering theorem and a distortion theorem, the Fekete-Szeg\"{o} inequality, lower bound for the Bloch constant, and Alexander's type theorem for such mappings.