# Hausdorff measure of escaping sets on certain meromorphic functions

## TL;DR

This paper investigates the Hausdorff measure of escaping sets for certain transcendental meromorphic functions, extending previous bounds to functions of infinite order by identifying gauge functions that determine measure zero or infinity.

## Contribution

It extends the analysis of Hausdorff measures of escaping sets to transcendental meromorphic functions of infinite order, providing new gauge functions for measure classification.

## Key findings

- Identifies gauge functions for infinite order functions.
- Determines conditions for Hausdorff measure to be zero or infinite.
- Extends previous bounds to a broader class of functions.

## Abstract

We consider transcendental meromorphic function for which the set of finite singularities of its inverse is bounded. Bergweiler and Kotus gave bounds for the Hausdorff dimension of escaping sets if the function has no logarithmic singularities over infinity, the multiplicities of poles are bounded and the order is finite. We study the case of infinite order and find gauge functions for which the Hausdorff measure of escaping sets is zero or infinity.