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Possible periods of Herman rings are studied for general meromorphic functions with at least one omitted value. A pole is called Hrelevant 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 ≥ 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 ≥ 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.

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Iteration of the function f λ (z) = λ + z + tan z, z ∈ C is investigated in this article.It is proved that for every λ, the Fatou set of f λ has a completely invariant Baker domain B; we call it the primary Fatou component. The rest of the results deals with f λ when it is topologically hyperbolic. For all real λ or λ such that λ = πk + iλ 2 for some integer k and 0 < λ 2 < 1, the only other Fatou component is shown to be another completely invariant Baker domain.It is proved that if |2 + λ 2 | < 1, then the Fatou set is the union of B and infinitely many invariant attracting domains. Every such domain U has exactly one invariant access to infinity and is unbounded in a special way;then it is found that the primary Fatou component is the only Fatou component and the Julia set is disconnected. For every natural number k, the Fatou set of f λ for λ = kπ + i π 2 is shown to contain k wandering domains with distinct grand orbits. These wandering domains are found to be escaping. The Fatou set is the union of B, these wandering domains and their pre-images.