Uniqueness theorems for meromorphic functions on annuli

A. Rathod

Ufa mathematical journal, Tome 12 (2020) no. 1, pp. 114-120

Résumé

In this paper, we discuss the uniqueness problems of meromorphic functions on annuli. We prove a general theorem on the uniqueness of meromorphic functions on annuli. An analogue of a famous Nevanlinna's five-value theorem is proposed. The main result in this paper is an analog of a result on the plane $\mathbb{C}$ obtained by H.S. Gopalkrishna and Subhas S. Bhoosnurmath for an annuli. That is, let $f_{1}(z)$ and $f_{2}(z)$ be two transcendental meromorphic functions on the annulus $\mathbb{A}=\left\{z:\frac{1}{R_{0}}|z|$, where $1$ Let $a_{j}$, $j=1,2,\ldots,q)$, be $q$ distinct complex numbers in $\overline{\mathbb{C}}$, and $k_{j}$, $j=1,2,\ldots,q$ be positive integers or $\infty$ satisfying \begin{equation*} k_{1}\geq k_{2}\geq \ldots \geq k_{q}. \end{equation*} If \begin{equation*} \overline{E}_{k_{j})}(a_{j},f_{1})=\overline{E}_{k_{j})}(a_{j},f_{2}), j=1,2,\ldots,q, \end{equation*} and \begin{equation*} \sum_{j=2}^{q}\frac{k_{j}}{k_{j}+1}-\frac{k_{1}}{k_{1}+1}>2, \end{equation*} then $f_{1}(z)\equiv f_{2}(z).$

Détail

Keywords: Nevanlinna theory, meromorphic functions
Mots-clés : annuli.