Variations on a theme of Hardy concerning the maximum modulus

Pardo‐Simón, Leticia and Sixsmith, David J. (2020). Variations on a theme of Hardy concerning the maximum modulus. Bulletin of the London Mathematical Society, 52(6) pp. 1134–1147.

DOI: https://doi.org/10.1112/blms.12387

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, $M(f)$ , has infinitely many discontinuities. This is one of only two known examples of such an entire 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 $M(f)$ 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.