Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A normality Criterion for a Family of Meromorphic Functions

Gopal Datt, Sanjay Kumar

Abstract

Schwick, in [6], states that let $\mathcal{F}$ be a family of meromorphic functions on a domain $D$ and if for each $f\in\mathcal{F}$, $(f^n)^{(k)}\neq 1$, for $z\in D$, where $n, k$ are positive integers such that $n\geq k+3$, then $\mathcal{F}$ is a normal family in $D$. In this paper, we investigate the opposite view that if for each $f\in\mathcal{F}$, $(f^n)^{(k)}(z)-ψ(z)$ has zeros in $D$, where $ψ(z)$ is a holomorphic function in $D$, then what can be said about the normality of the family $\mathcal{F}$?

A normality Criterion for a Family of Meromorphic Functions

Abstract

Schwick, in [6], states that let be a family of meromorphic functions on a domain and if for each , , for , where are positive integers such that , then is a normal family in . In this paper, we investigate the opposite view that if for each , has zeros in , where is a holomorphic function in , then what can be said about the normality of the family ?

Paper Structure

This paper contains 4 sections, 9 theorems, 36 equations.

Table of Contents

  1. Introduction and main results
  2. Some Notations
  3. Preliminary results
  4. Proof of Theorem \ref{['theorem 1']}

Key Result

Theorem 1.1

Let $a_1, a_2,\ldots, a_q,$ be $q$ distinct non-zero complex values and $l_1, l_2,\ldots, l_q$ be $q$ positive integers $($or $+\infty$$)$, where $q\geq1$. Let $n$ be a non-negative integer, and $n_1,\ldots, n_k, t_1,\ldots, t_k$ positive integers $(k\geq1).$ Let $\mathcal{F}$ be a family of meromor Then $\mathcal{F}$ is normal in $D$.

Theorems & Definitions (11)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Lemma 3.1: Zalcman's lemma
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • proof
  • Theorem 4.1
  • ...and 1 more