Iterating the Minimum Modulus: Functions of Order Half, Minimal Type

Nicks, D. A.; Rippon, P. J.; Stallard, G. M.

Abstract

For a transcendental entire functionf, the property that there exists r> 0 such that mn(r) → ∞ as n→ ∞, where m(r) = min { | f(z) | : | z| = r} , is related to conjectures of Eremenko and of Baker, for both of which order 1/2 minimal type is a significant rate of growth. We show that this property holds for functions of order1/2 minimal type if the maximum modulus off has sufficiently regular growth and we give examples to show the sharpness of our results by using a recent generalisation of Kjellberg’s method of constructing entire functions of small growth, which allows rather precise control of m(r).

Citation

Nicks, D. A., Rippon, P. J., & Stallard, G. M. (2021). Iterating the Minimum Modulus: Functions of Order Half, Minimal Type. Computational Methods and Function Theory, 21(4), 653–670. https://doi.org/10.1007/s40315-021-00400-w

• Journal Article Type: Article
• Acceptance Date: Mar 17, 2021
• Online Publication Date: Jul 13, 2021
• Publication Date: 2021-12
• Deposit Date: Jul 26, 2021
• Publicly Available Date: Jul 26, 2021
• Journal: Computational Methods and Function Theory
• Print ISSN: 1617-9447
• Electronic ISSN: 2195-3724
• Publisher: Springer Verlag
• Peer Reviewed: Peer Reviewed
• Volume: 21
• Issue: 4
• Pages: 653–670
• DOI: https://doi.org/10.1007/s40315-021-00400-w
• Keywords: Applied Mathematics; Computational Theory and Mathematics; Analysis
• Public URL: https://nottingham-repository.worktribe.com/output/5833580
• Publisher URL: https://link.springer.com/article/10.1007/s40315-021-00400-w