Herman rings of meromorphic maps with an omitted value

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.