Abstract

Let $ S=\{z\in {\mathbb C}:P(z)=z^n+ az^{n-1}+ b =0 \}$, where $ a, b \in \mathbb C $ be nonzero constants satisfying $ \dfrac{b}{a^n} \ne \dfrac{(-1)^n(n-1)^{n-1}}{n^{n}}$. The uniqueness of meromorphic functions sharing $ S $ counting multiplicity(resp. with weight $ 2 $) has been studied by Yi ([18]) (resp. Lahiri, Banerjee ([12])). In this paper, we consider the uniqueness of meromorphic functions sharing $ S $ ignoring multiplicity. We first obtain the analog of Yi's Theorem 2 ([18]). Next, we show that $ S $ is a unique range set for the class of meromorphic functions ignoring multiplicity of higher multiplicities of either zeros or poles, which different from S. Mallick - D. Sarkar's ([13]). We discuss some applications of the main result. Our results are inspired by a work of Yi ([18]) and Khoai ([11]).

Keywords and Phrases

Uniqueness, ignoring multiplicity, multiplicities of zeros, poles of meromorphic functions.

A.M.S. subject classification

30D35.

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