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arXiv · 1911.05408

Variations on a theme of Hardy concerning the maximum modulus

L. Pardo-Simón·D. J. Sixsmith
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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.

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L. Pardo-Simón, D. J. Sixsmith. 2019-11-13. Variations on a theme of Hardy concerning the maximum modulus. https://doi.org/10.1112/blms.12387

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