Hyperbolic metrics, homogeneous holomorphic functionals and Zalcman's conjecture

TL;DR

This paper demonstrates that certain holomorphic functionals are maximized by the Koebe function using hyperbolic metrics, leading to proofs of the Zalcman and Bieberbach conjectures.

Contribution

It introduces a hyperbolic metric approach to identify extremal functions in univalent function theory, providing new proofs of longstanding conjectures.

Findings

Holomorphic functionals are maximized by the Koebe function.

Hyperbolic metrics determine extremality in univalent functions.

Proofs of Zalcman and Bieberbach conjectures are established.

Abstract

We show, using the Kobayashi and Caratheodory metrics on special holomorphic disks in the universal Teichmuller space, that a wide class of holomorphic functionals on the space of univalent functions in the disk is maximized by the Koebe function or by its root transforms; their extremality is forced by hyperbolic features. As consequences, this implies the proofs of the famous Zalcman and Bieberbach conjectures.