Herman rings of meromorphic maps with an omitted value

TL;DR

This paper studies the conditions under which Herman rings occur in transcendental meromorphic functions with omitted values, revealing restrictions based on pole multiplicity and period, and characterizing certain Fatou components.

Contribution

It provides new non-existence results for Herman rings in specific classes of meromorphic functions with omitted values and characterizes doubly connected periodic Fatou components.

Findings

Functions with all multiple poles have no Herman rings.

Herman rings of period one or two do not exist.

Doubly connected periodic Fatou components are Herman rings.

Abstract

We investigate the existence and distribution of Herman rings of transcendental meromorphic functions which have at least one omitted value. If all the poles of such a function are multiple then it has no Herman ring. Herman rings of period one or two do not exist. Functions with a single pole or with at least two poles one of which is an omitted value have no Herman ring. Every doubly connected periodic Fatou component is a Herman ring.