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Maximum and average valence of meromorphic functions

Aimo Hinkkanen, Joseph Miles

Abstract

If $f$ is a meromorphic function from the complex plane ${\mathbb C}$ to the extended complex plane $\overline{ {\mathbb C} }$, for $r > 0$ let $n(r)$ be the maximum number of solutions in $\{z\colon |z| \leq r \}$ of $f(z) = a$ for $a \in \overline{ {\mathbb C} }$, and let $A(r,f)$ be the average number of such solutions. Using a technique introduced by Toppila, we exhibit a meromorphic function for which $\liminf_{r\to\infty} n(r)/A(r,f) \geq 1.07328$.

Maximum and average valence of meromorphic functions

Abstract

If is a meromorphic function from the complex plane to the extended complex plane , for let be the maximum number of solutions in of for , and let be the average number of such solutions. Using a technique introduced by Toppila, we exhibit a meromorphic function for which .
Paper Structure (10 sections, 1 theorem, 192 equations)

This paper contains 10 sections, 1 theorem, 192 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Construction of the Example
  4. The function $f$
  5. A sketch of the proof
  6. Certain circles and annuli
  7. Certain image areas
  8. Intersections of certain circles
  9. Bounds for image areas
  10. The final estimates

Key Result

Theorem 1.1

There exists a non-constant meromorphic function $f$ in the plane for which

Theorems & Definitions (1)

  • Theorem 1.1