On periods of Herman rings and relevant poles

TL;DR

This paper investigates the possible periods of Herman rings in meromorphic functions with omitted values, establishing lower bounds based on relevant poles and exploring conditions for the existence or absence of Herman rings.

Contribution

It provides new lower bounds on the period of Herman rings in terms of relevant poles and identifies conditions for their existence or non-existence.

Findings

Lower bound p ≥ h(h+1)/2 when Herman ring surrounds a pole and omitted values.

Lower bound p ≥ h(h+3)/2 in other cases.

Herman rings do not exist if an omitted value lies in the closure of certain Fatou components.

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.