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ZEROS OF DIFFERENTIAL POLYNOMIALS IN REAL MEROMORPHIC FUNCTIONS

Published online by Cambridge University Press:  23 May 2005

Walter Bergweiler ,
Alex Eremenko  and
Jim K. Langley
Walter Bergweiler
Affiliation:
Mathematisches Seminar, Christian–Albrechts–Universität zu Kiel, Ludewig–Meyn–Str. 4, D-24098 Kiel, Germany (bergweiler@math.uni-kiel.de)
Alex Eremenko
Affiliation:
Purdue University, West Lafayette, IN 47907, USA (eremenko@math.purdue.edu)
Jim K. Langley
Affiliation:
School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK (jkl@maths.nottingham.ac.uk)
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Abstract

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We investigate whether differential polynomials in real transcendental meromorphic functions have non-real zeros. For example, we show that if $g$ is a real transcendental meromorphic function, $c\in\mathbb{R}\setminus\{0\}$ and $n\geq3$ is an integer, then $g'g^n-c$ has infinitely many non-real zeros. If $g$ has only finitely many poles, then this holds for $n\geq2$. Related results for rational functions $g$ are also considered.

Keywords

Primary 30D35meromorphic functiondifferential polynomialreal zeros
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 48 , Issue 2 , June 2005 , pp. 279 - 293
Copyright
Copyright © Edinburgh Mathematical Society 2005