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A normality criterion for a family of meromorphic functions

Kuntal Mandal, Bipul Pal

Abstract

We consider a family $\mathscr{F}$ of meromorphic functions defined in a domain $D$, a holomorphic function $ψ$ and a homogeneous differential polynomial $ P[f] $ of degree $d$ with weight $w$. In this paper, we prove the normality of $\mathscr{F}$ under certain conditions such as $f\neq 0$, $P[f]\neq 0$ and all the zeros of the function $P[f] - ψ^d$ have multipicity at least $\displaystyle{\frac{w+1}{w-1}}$, for each $f \in \mathscr{F}$.

A normality criterion for a family of meromorphic functions

Abstract

We consider a family of meromorphic functions defined in a domain , a holomorphic function and a homogeneous differential polynomial of degree with weight . In this paper, we prove the normality of under certain conditions such as , and all the zeros of the function have multipicity at least , for each .
Paper Structure (3 sections, 4 theorems, 58 equations)

This paper contains 3 sections, 4 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Lemmas
  3. Proof the theorem \ref{['m']}

Key Result

Theorem 1.1

Let $\mathscr{F}$ be a family of meromorphic functions defined in a domain $D \subset \mathbb{C}$, let $\psi (\not\equiv 0)$ and $a_1, a_2, \ldots, a_{n}$ be holomorphic functions in $D$, and let $k$, $n$ be two positive integers. Suppose $P[f]$ be a homogeneous differential polynomial of degree $d$

Theorems & Definitions (7)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • proof