On periods of Herman rings and relevant poles

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 $$p\ge \frac{h(h+1)}{2}$$ p ≥ h ( h + 1 ) 2 whenever $$f^j(H)$$ f j ( H ) , for some j, surrounds a pole as well as the set of all omitted values of f. It is proved that $$p \ge \frac{h(h+3)}{2}$$ p ≥ 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.