Bank–Laine functions, the Liouville transformation and the Eremenko–Lyubich class

Langley, James K.

Abstract

The Bank–Laine conjecture concerning the oscillation of solutions of second order homogeneous linear differential equations has recently been disproved by Bergweiler and Eremenko. It is shown here, however, that the conjecture is true if the set of finite critical and asymptotic values of the coefficient function is bounded. It is also shown that if E is a Bank–Laine function of finite order with infinitely many zeros, all real and positive, then its zeros must have exponent of convergence at least 3/2, and an example is constructed via quasiconformal surgery to demonstrate that this result is sharp.

Citation

Langley, J. K. (2020). Bank–Laine functions, the Liouville transformation and the Eremenko–Lyubich class. Journal d'Analyse Mathématique, 141(1), 225-246. https://doi.org/10.1007/s11854-020-0115-6

• Journal Article Type: Article
• Acceptance Date: Dec 14, 2018
• Online Publication Date: Nov 12, 2020
• Publication Date: 2020-09
• Deposit Date: Jan 3, 2019
• Publicly Available Date: Oct 1, 2021
• Journal: Journal d'Analyse Mathématique
• Print ISSN: 0021-7670
• Electronic ISSN: 1565-8538
• Peer Reviewed: Peer Reviewed
• Volume: 141
• Issue: 1
• Pages: 225-246
• DOI: https://doi.org/10.1007/s11854-020-0115-6
• Public URL: https://nottingham-repository.worktribe.com/output/1441685
• Publisher URL: https://link.springer.com/article/10.1007/s11854-020-0115-6
• Additional Information: Received: 16 October 2018; Revised: 30 November 2018; First Online: 12 November 2020