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A uniqueness problem concerning entire functions and their derivatives

Andreas Sauer, Andreas Schweizer

Abstract

We determine all entire functions $f$ such that for nonzero complex values $a\neq b$ the implications $f=a \Rightarrow f' =a$ and $f' =b \Rightarrow f=b$ hold. This solves an open problem in uniqueness theory. In this context we give a normality criterion, which might be interesting in its own right.

A uniqueness problem concerning entire functions and their derivatives

Abstract

We determine all entire functions such that for nonzero complex values the implications and hold. This solves an open problem in uniqueness theory. In this context we give a normality criterion, which might be interesting in its own right.
Paper Structure (7 sections, 28 theorems, 77 equations)

This paper contains 7 sections, 28 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. A normality criterion
  3. A uniqueness problem
  4. Periodicity
  5. Polynomials in $\exp(\lambda z)$
  6. The $b$-points of $f'$
  7. The end of the proof

Key Result

Theorem 1.1

(Main Theorem) Let $f$ be a nonconstant entire function and let $a\neq b$ be two nonzero complex numbers. If and for all $z_0 \in \mathbb C$, then $f$ is one of the following:

Theorems & Definitions (34)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Theorem 2.2
  • Example 2.3
  • Definition 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Theorem 3.5
  • ...and 24 more