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THE SECOND DERIVATIVE OF A MEROMORPHIC FUNCTION

Published online by Cambridge University Press:  20 January 2009

J. K. Langley
Affiliation:
School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK ([email protected])
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Abstract

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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

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2001