Meromorphic functions sharing three values with their shift

Sujoy Majumder; Pradip Das

doi:10.3842/umzh.v76i5.7502

Sujoy Majumder Department of Mathematics, Raiganj University, West Bengal, India
Pradip Das Department of Mathematics, Raiganj University, West Bengal, India

DOI: https://doi.org/10.3842/umzh.v76i5.7502

Keywords: meromorphic function; uniqueness theory; shared values; Nevanlinna theory; shift; difference.

Abstract

UDC 517.5

We discuss the problem of uniqueness of a meromorphic function $f(z),$ which shares $a_1(z)$, $a_2(z),$ and $a_3(z)$ CM with its shift $f(z+c)$, where $a_1(z)$, $a_2(z),$ and $a_3(z)$ are three $c$-periodic distinct small functions of $f(z)$ and $c\in\mathbb{C}\setminus\{0\}$. The obtained result improves the recent result of Heittokangas et al. [Complex Var. and Elliptic Equat., 56, No. 1–4, 81–92 (2011)] by dropping the assumption about the order of $f(z)$. In addition, we introduce a way of characterizing elliptic functions in terms of meromorphic functions sharing values with two of their shifts. Moreover, we show by a number of illustrating examples that our results are, in certain senses, best possible.

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