Baker's conjecture for functions with real zeros

Nicks, Daniel A.; Rippon, Philip J. and Stallard, Gwyneth M. (2018). Baker's conjecture for functions with real zeros. Proceedings of the London Mathematical Society, 117(1) pp. 100–124.

DOI: https://doi.org/10.1112/plms.12124

Abstract

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.

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About

  • Item ORO ID
  • 53677
  • Item Type
  • Journal Item
  • 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 or School
  • Faculty of Science, Technology, Engineering and Mathematics (STEM) > Mathematics and Statistics
    Faculty of Science, Technology, Engineering and Mathematics (STEM)
  • Copyright Holders
  • © 2018 London Mathematical Society
  • Depositing User
  • Philip Rippon

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