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A remark concerning normal families and shared values

Andreas Sauer

Abstract

We improve well-known results concerning normal families and shared values of meromorphic functions in the plane. In particular, we obtain two corollaries concerning meromorphic functions $f \colon {\mathbb C} \to {\widehat{\mathbb C}}$: i) If $f$ shares a non-zero finite value with $f'$, and such that $f'$ is bounded on the preimages of $f$ for a second value, then $f$ is normal. ii) If $f$ shares two finite values with $f'$, then $f$ and $f'$ are normal.

A remark concerning normal families and shared values

Abstract

We improve well-known results concerning normal families and shared values of meromorphic functions in the plane. In particular, we obtain two corollaries concerning meromorphic functions : i) If shares a non-zero finite value with , and such that is bounded on the preimages of for a second value, then is normal. ii) If shares two finite values with , then and are normal.

Paper Structure

This paper contains 3 sections, 7 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. One non-zero shared value and one value with bounded derivative
  3. Two shared values and derivatives

Key Result

Theorem 2.1

Let $\cal F$ be a family of meromorphic functions on a domain $D \subset \mathbb C$, $a \neq 0$ and $b \neq 0$ be complex numbers and $K \ge 1$. If for every $f \in \cal F$ and all $z \in D$ then $\cal F$ is a normal family.

Theorems & Definitions (9)

  • Theorem 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Example 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Theorem 3.1
  • Corollary 3.2
  • Example 3.3