Herman rings with small periods and omitted values

Abstract

All possible arrangements of cycles of three periodic as well as four periodic Herman rings of transcendental meromorphic functions having at least one omitted value are determined. It is shown that if p = 3 or 4, then the number of p-cycles of Herman rings is at most one. We have also proved a result about the non-existence of a 3-cycle and a 4-cycle of Herman rings simultaneously. Finally some examples of functions having no Herman ring are discussed.