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On a Theorem of Schwick

Gopal Datt, Sanjay Kumar

Abstract

Let D be a domain, n, k be positive integers and n >= K+3. Let F be a family of functions meromorphic in D. If each f in F satisfies (f^n)^(k) not equal to 1 for z in D, then F is normal family. This result was proved by Schwick. In this paper we extend this result.

On a Theorem of Schwick

Abstract

Let D be a domain, n, k be positive integers and n >= K+3. Let F be a family of functions meromorphic in D. If each f in F satisfies (f^n)^(k) not equal to 1 for z in D, then F is normal family. This result was proved by Schwick. In this paper we extend this result.

Paper Structure

This paper contains 4 sections, 15 theorems, 26 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminary results
  3. Proof of Theorem \ref{['thm']}
  4. Proof of Theorem \ref{['thm1']}

Key Result

Theorem 1.1

Let $n, k$ be positive integers and $n\geq k+1$ and $\mathcal{D}$ be a domain in $\mathbb{C}$. Let $\mathcal{F}$ be a family of functions meromorphic on $\mathcal{D}$. If each $f\in \mathcal{F}$ satisfies $(f^n)^{(k)}(z)\neq 1$ for $z\in \mathcal{D}$, then $\mathcal{F}$ is a normal family.

Theorems & Definitions (18)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • ...and 8 more