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ZEROS OF DIFFERENTIAL POLYNOMIALS IN REAL MEROMORPHIC FUNCTIONS

Published online by Cambridge University Press:  23 May 2005

Walter Bergweiler
Affiliation:
Mathematisches Seminar, Christian–Albrechts–Universität zu Kiel, Ludewig–Meyn–Str. 4, D-24098 Kiel, Germany ([email protected])
Alex Eremenko
Affiliation:
Purdue University, West Lafayette, IN 47907, USA ([email protected])
Jim 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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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.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2005