A lower bound of the hyperbolic dimension for meromorphic functions having a logarithmic H\"older tract

TL;DR

This paper establishes improved lower bounds for the hyperbolic dimension of certain meromorphic functions with logarithmic H"older tracts, linking these bounds to the fractal nature of the boundary at infinity.

Contribution

It introduces new lower bounds for hyperbolic dimension based on the fractal properties of the boundary of logarithmic H"older tracts in meromorphic functions.

Findings

Enhanced lower bounds for hyperbolic dimension.

Connection between fractal boundary behavior and hyperbolic dimension.

Quantitative estimates using integral means at infinity.

Abstract

We improve existing lower bounds of the hyperbolic dimension for meromophic functions that have a logarithmic tract {\Omega} which is a H\"older domain. These bounds are given in terms of the fractal behavior, measured with integral means, of the boundary of {\Omega} at infinity.