Baker's conjecture for functions with real zeros

Nicks, Daniel A. and Rippon, P.J. and Stallard, G.M. (2018) Baker's conjecture for functions with real zeros. Proceedings of the London Mathematical Society . ISSN 1460-244X (In Press)

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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.

Item Type: Article
Keywords: Entire function; Baker's conjecture; Unbounded wandering domain; Real zeros; Minimum modulus; Winding of image curves; Extremal length; Laguerre-Pólya class
Schools/Departments: University of Nottingham, UK > Faculty of Science > School of Mathematical Sciences
Depositing User: Eprints, Support
Date Deposited: 01 Mar 2018 09:34
Last Modified: 10 Mar 2018 06:45

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