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 $p\geq \frac{h(h+1)}{2}$ 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 $p \geq \frac{h(h+3)}{2}$ 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.