Baker's conjecture for functions with real zeros
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...
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London Mathematical Society
2018
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| Online Access: | https://eprints.nottingham.ac.uk/50092/ |
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| author | Nicks, Daniel A. Rippon, P.J. Stallard, G.M. |
| author_facet | Nicks, Daniel A. Rippon, P.J. Stallard, G.M. |
| author_sort | Nicks, Daniel A. |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | 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. |
| first_indexed | 2025-11-14T20:15:12Z |
| format | Article |
| id | nottingham-50092 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T20:15:12Z |
| publishDate | 2018 |
| publisher | London Mathematical Society |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-500922020-05-04T19:47:46Z https://eprints.nottingham.ac.uk/50092/ Baker's conjecture for functions with real zeros Nicks, Daniel A. Rippon, P.J. Stallard, G.M. 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. London Mathematical Society 2018-07-31 Article PeerReviewed Nicks, Daniel A., Rippon, P.J. and Stallard, G.M. (2018) Baker's conjecture for functions with real zeros. Proceedings of the London Mathematical Society, 117 (1). pp. 100-124. ISSN 1460-244X Entire function; Baker's conjecture; Unbounded wandering domain; Real zeros; Minimum modulus; Winding of image curves; Extremal length; Laguerre-Pólya class https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/plms.12124 doi:10.1112/plms.12124 doi:10.1112/plms.12124 |
| spellingShingle | Entire function; Baker's conjecture; Unbounded wandering domain; Real zeros; Minimum modulus; Winding of image curves; Extremal length; Laguerre-Pólya class Nicks, Daniel A. Rippon, P.J. Stallard, G.M. Baker's conjecture for functions with real zeros |
| title | Baker's conjecture for functions with real zeros |
| title_full | Baker's conjecture for functions with real zeros |
| title_fullStr | Baker's conjecture for functions with real zeros |
| title_full_unstemmed | Baker's conjecture for functions with real zeros |
| title_short | Baker's conjecture for functions with real zeros |
| title_sort | baker's conjecture for functions with real zeros |
| topic | Entire function; Baker's conjecture; Unbounded wandering domain; Real zeros; Minimum modulus; Winding of image curves; Extremal length; Laguerre-Pólya class |
| url | https://eprints.nottingham.ac.uk/50092/ https://eprints.nottingham.ac.uk/50092/ https://eprints.nottingham.ac.uk/50092/ |