Variations on a theme of Hardy concerning the maximum modulus

TL;DR

This paper generalizes Hardy’s classical example by constructing transcendental entire functions with maximum modulus discontinuities at prescribed sequences, including functions of finite order within the Eremenko-Lyubich class.

Contribution

It introduces a method to construct transcendental entire functions with maximum modulus discontinuities at any given increasing sequence, extending Hardy's original example.

Findings

Constructed functions with prescribed discontinuities in maximum modulus

Included functions of finite order within the Eremenko-Lyubich class

Extended the class of known examples of such functions

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.