Variations on a theme of Hardy concerning the maximum modulus

Abstract

In 1909, Hardy gave an example of a transcendental entire function, $f$, with the property that the set of points where $f$ achieves its maximum modulus, $\mathcal{M}(f)$, has infinitely many discontinuities. This is one of only two known examples of such a function. In this paper we significantly generalise these examples. In particular, we show that, given an increasing sequence of positive real numbers, tending to infinity, there is a transcendental entire function, $f$, such that $\mathcal{M}(f)$ has discontinuities with moduli at all these values. We also show that the transcendental entire function lies in the much studied Eremenko-Lyubich class. Finally, we show that, with an additional hypothesis on the sequence, we can ensure that $f$ has finite order.

arXiv commentAuthor accepted manuscript. To appear in Bull. London Math. Soc

References in corpus (4)

• Rigidity of escaping dynamics for transcendental entire functions
• Hyperbolic entire functions with full hyperbolic dimension and approximation by Eremenko-Lyubich functions
• Maximally and non-maximally fast escaping points of transcendental entire functions
• Hyperbolic entire functions and the Eremenko-Lyubich class: Class $\mathcal{B}$ or not class $\mathcal{B}$?

Cited by in corpus (1)

• The maximum modulus set of a polynomial