Boundary distortion estimates for holomorphic maps

TL;DR

This paper derives new boundary distortion estimates for holomorphic self-maps of the unit disk, extending classical results by employing extremal length and a novel semigroup approach, applicable beyond univalent functions.

Contribution

It introduces a new method combining extremal length and semigroup theory to estimate boundary derivatives of holomorphic maps, including non-univalent functions.

Findings

Established lower bounds for angular derivatives at boundary fixed points.

Extended classical univalent function estimates to broader classes of holomorphic maps.

Derived new inequalities for maps with images not separating the origin and boundary.

Abstract

We establish some estimates of the the angular derivatives from below for holomorphic self-maps of the unit disk at one and two fixed points of the unit circle provided there is no fixed point inside the unit disk. The results complement Cowen-Pommerenke and Anderson-Vasil'ev type estimates in the case of univalent functions. We use the method of extremal length and propose a new semigroup approach to deriving inequalities for holomorphic self-maps of the disk which are not necessarily univalent using known inequalities for univalent functions. This approach allowed us to receive a new Ossermans type estimate as well as inequalities for holomorphic self-maps which images do not separate the origin and the boundary.