Non-existence of unbounded Fatou components of a meromorphic function

TL;DR

This paper establishes conditions under which transcendental meromorphic functions lack unbounded Fatou components, extending previous results from entire functions and exploring the non-existence of unbounded wandering domains.

Contribution

It provides new sufficient conditions for the non-existence of unbounded Fatou components in transcendental meromorphic functions, including compositions of such functions.

Findings

Conditions for non-existence of unbounded Fatou components

Extension of results from entire to meromorphic functions

Analysis of unbounded wandering domains

Abstract

This paper is devoted to establish sufficient conditions under which a transcendental meromorphic function has no unbounded Fatou components and to extend some results for entire functions to meromorphic functions. Actually, we shall mainly discuss non-existence of unbounded wandering domains of a meromorphic function. The case for a composition of finitely many meromorphic function with at least one of them being transcendental can be also investigated in the argument of this paper.