Baker's conjecture for functions with real zeros

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

Abstract

Baker's conjecture states that a transcendental entire function of order less than 1=2 has no unbounded Fatou components. It is known that, for such functions, there are no unbounded periodic Fatou components and so it remains to show that they can also have no unbounded wandering domains. Here we introduce completely new techniques to show that the conjecture holds in the case that the transcendental entire function is real with only real zeros, and we prove the much stronger result that such a function has no orbits consisting of unbounded wandering domains whenever the order is less than 1. This raises the question as to whether such wandering domains can exist for any transcendental entire function with order less than 1. Key ingredients of our proofs are new results in classical complex analysis with wider applications. These new results concern: the winding properties of the images of certain curves proved using extremal length arguments, growth estimates for entire functions, and the distribution of the zeros of entire functions of order less than 1.

Citation

Nicks, D. A., Rippon, P., & Stallard, G. (2018). Baker's conjecture for functions with real zeros. Proceedings of the London Mathematical Society, 117(1), 100-124. https://doi.org/10.1112/plms.12124

• Journal Article Type: Article
• Acceptance Date: Feb 16, 2018
• Online Publication Date: Mar 31, 2018
• Publication Date: Jul 31, 2018
• Deposit Date: Mar 1, 2018
• Publicly Available Date: Mar 31, 2018
• Journal: Proceedings of the London Mathematical Society
• Print ISSN: 0024-6115
• Electronic ISSN: 1460-244X
• Publisher: London Mathematical Society
• Peer Reviewed: Peer Reviewed
• Volume: 117
• Issue: 1
• Pages: 100-124
• DOI: https://doi.org/10.1112/plms.12124
• Keywords: Entire function; Baker's conjecture; Unbounded wandering domain; Real zeros; Minimum modulus; Winding of image curves; Extremal length; Laguerre-Pólya class
• Public URL: https://nottingham-repository.worktribe.com/output/948299
• Publisher URL: https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/plms.12124
• Additional Information: Copyright © 1999-2018 John Wiley & Sons, Inc. All rights reserved. This is the ; accepted version of the following article: Nicks, D. A., Rippon, P. J. and Stallard, G. M. (2018), Baker's conjecture for functions with real zeros. Proc. London Math. Soc., 117: 100-124. doi:10.1112/plms.12124, which has been published in final form at https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/plms.12124