Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-26T18:14:21.683Z Has data issue: false hasContentIssue false
  • English
  • Français

THE SECOND DERIVATIVE OF A MEROMORPHIC FUNCTION

Published online by Cambridge University Press:  20 January 2009

J. K. Langley
J. K. Langley
Affiliation:
School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK ([email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let $f$ be meromorphic of finite order in the plane, such that $f^{(k)}$ has finitely many zeros, for some $k\geq2$. The author has conjectured that $f$ then has finitely many poles. In this paper, we strengthen a previous estimate for the frequency of distinct poles of $f$. Further, we show that the conjecture is true if either

  1. $f$ has order less than $1+\varepsilon$, for some positive absolute constant $\varepsilon$, or

  2. $f^{(m)}$, for some $0\leq m lt k$, has few zeros away from the real axis.

AMS 2000 Mathematics subject classification: Primary 30D35

Keywords

meromorphic functionNevanlinna theoryzeros of derivatives
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 44 , Issue 3 , October 2001 , pp. 455 - 478
Copyright
Copyright © Edinburgh Mathematical Society 2001