Eremenko's Conjecture for Functions with Real Zeros: The Role of the Minimum Modulus

Nicks, Daniel; Rippon, Philip and Stallard, Gwyneth (2021). Eremenko's Conjecture for Functions with Real Zeros: The Role of the Minimum Modulus. International Mathematics Research Notices, 2021(18) pp. 13946–13974.

DOI: https://doi.org/10.1093/imrn/rnaa020

Abstract

We consider the class of real transcendental entire functions of finite order with only real zeros, and show that if the iterated minimum modulus tends to , then the escaping set of has the structure of a spider's web, in which case Eremenko's conjecture holds. This minimum modulus condition is much weaker than that used in previous work on Eremenko's conjecture. For functions in this class we analyse the possible behaviours of the iterated minimum modulus in relation to the order of the function .

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About

• Item ORO ID
• 69153
• Item Type
• Journal Item
• ISSN
• 1687-0247
• Project Funding Details

• Funded Project Name	Project ID	Funding Body
  Baker's conjecture and Eremenko's conjecture: a unified approach (XM-08-066-GS)	EP/H006591/1	EPSRC (Engineering and Physical Sciences Research Council)

• Keywords
• transcendental entire function; escaping set; Eremenko's conjecture; spider's web; minimum modulus; order; genus; deficiency
• Academic Unit or School
• Faculty of Science, Technology, Engineering and Mathematics (STEM) > Mathematics and Statistics
  Faculty of Science, Technology, Engineering and Mathematics (STEM)
• Copyright Holders
• © 2019 D A Nicks, © 2019 P J Rippon, © 2019 G M Stallard
• Related URLs
• • https://arxiv.org/abs/1810.07814(Other)
• Depositing User
• Philip Rippon