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Baker's conjecture for functions with real zeros

Nicks, Daniel A.; Rippon, Philip J. and Stallard, Gwyneth M. (2019). Baker's conjecture for functions with real zeros. Proceedings of the London Mathematical Society (Early Access).

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Baker's conjecture states that a transcendental entire functions 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.

Item Type: Journal Item
Copyright Holders: 2018 London Mathematical Society
ISSN: 1460-244X
Project Funding Details:
Funded Project NameProject IDFunding Body
Bakers Conjecture and Eremenko's Conjecture: New Directions (XM-12-066-GS)EP/K031163/1EPSRC (Engineering and Physical Sciences Research Council)
Baker's conjecture and Eremenko's conjecture: a unified approach (XM-08-066-GS)EP/H006591/1EPSRC (Engineering and Physical Sciences Research Council)
Keywords: entire function; Baker’s conjecture; unbounded wandering domain; real zeros; minimum modulus; winding of image curves; extremal length; Laguerre–Polya class
Academic Unit/School: Faculty of Science, Technology, Engineering and Mathematics (STEM) > Mathematics and Statistics
Faculty of Science, Technology, Engineering and Mathematics (STEM)
Item ID: 53677
Depositing User: Philip Rippon
Date Deposited: 28 Feb 2018 09:50
Last Modified: 27 Jun 2018 08:37
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