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

Nicks, D. A.; Rippon, P. J.; Stallard, G. M.

Abstract

We consider the class of real transcendental entire functions f of finite order with only real zeros and show that if the iterated minimum modulus tends to 8, then the escaping set I(f) of f 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 f.

Citation

Nicks, D. A., Rippon, P. J., & Stallard, G. M. (2021). Eremenko's Conjecture for Functions with Real Zeros: The Role of the Minimum Modulus. International Mathematics Research Notices, 2021(18), 13946-13974. https://doi.org/10.1093/imrn/rnaa020

• Journal Article Type: Article
• Acceptance Date: Mar 13, 2020
• Online Publication Date: Apr 11, 2020
• Publication Date: Sep 1, 2021
• Deposit Date: Mar 17, 2020
• Publicly Available Date: Apr 12, 2021
• Journal: International Mathematics Research Notices
• Print ISSN: 1073-7928
• Electronic ISSN: 1687-0247
• Publisher: Oxford University Press
• Peer Reviewed: Peer Reviewed
• Volume: 2021
• Issue: 18
• Article Number: rnaa020
• Pages: 13946-13974
• DOI: https://doi.org/10.1093/imrn/rnaa020
• Keywords: General Mathematics
• Public URL: https://nottingham-repository.worktribe.com/output/4160199
• Publisher URL: https://academic.oup.com/imrn/article-abstract/doi/10.1093/imrn/rnaa020/5818934
• Additional Information: This is a pre-copyedited, author-produced version of an article accepted for publication in International Mathematics Research Notices following peer review. The version of record D A Nicks, P J Rippon, G M Stallard, Eremenko’s Conjecture for Functions with Real Zeros: The Role of the Minimum Modulus, International Mathematics Research Notices, Volume 2021, Issue 18, September 2021, Pages 13946–13974, https://doi.org/10.1093/imrn/rnaa020 is available online at: https://academic.oup.com/imrn/article-abstract/doi/10.1093/imrn/rnaa020/5818934.