On periods of Herman rings and relevant poles

Abstract

Possible periods of Herman rings are studied for general meromorphic functions with at least one omitted value. A pole is called H-relevant for a Herman ring H of such a function f if it is surrounded by some Herman ring of the cycle containing H. In this article, a lower bound on the period p of a Herman ring H is found in terms of the number of H-relevant poles, say h. More precisely, it is shown that \(p\ge \frac{h(h+1)}{2}\) whenever \(f^j(H)\), for some j, surrounds a pole as well as the set of all omitted values of f. It is proved that \(p \ge \frac{h(h+3)}{2}\) in the other situation. Sufficient conditions are found under which equalities hold. It is also proved that if an omitted value is contained in the closure of an invariant or a two periodic Fatou component then the function does not have any Herman ring.