Entire Functions with Separated Zeros and 1-Points
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Entire Functions with Separated Zeros and 1-Points Walter Bergweiler1 · Alexandre Eremenko2 Received: 2 February 2020 / Accepted: 20 August 2020 © The Author(s) 2020
Abstract We consider transcendental entire functions of finite order for which the zeros and 1-points are in disjoint sectors. Under suitable hypotheses on the sizes of these sectors we show that such functions must have a specific form, or that such functions do not exist at all. Keywords Entire function · Value distribution · Sector · Radially distributed value · Subharmonic function Mathematics Subject Classification 30D20 · 30D35
1 Introduction and Results Our starting point is the following result of Biernacki [3, p. 533]. Theorem A There is no transcendental entire function of finite order for which the zeros accumulate in one direction and the 1-points accumulate in a different direction. Here we say that a set {an } of complex numbers accumulates in one direction if there exists a ray such that for every open sector bisected by this ray all but finitely many an lie in this sector.
Dedicated to the memory of Stephan Ruscheweyh Communicated by Filippo Bracci. Supported by NSF Grant DMS-1665115.
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Walter Bergweiler [email protected] Alexandre Eremenko [email protected]
1
Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Str. 4, 24098 Kiel, Germany
2
Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA
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W. Bergweiler, A. Eremenko
We will prove the following generalizations of Theorem A. Theorem 1.1 Let S0 and S1 be closed sectors in C satisfying S0 ∩ S1 = {0}. Let θ j denote the opening angle of S j and suppose that min{θ0 , θ1 }
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