Affiliation:
1. Department of Mathematics, University of Reading, Whiteknights, PO Box 220, Reading RG6 6AX, UK
Abstract
Abstract
In this paper, we study entire functions whose maximum on a disc of radius $r$ grows like $e^{h(\log r)}$ for some function $h(\cdot )$. We show that this is impossible if $h^{\prime \prime }(r)$ tends to a limit as $r\to \infty $, thereby solving a problem of Hayman from 1966. On the other hand, we show that entire functions can, under some mild smoothness conditions, grow like $\textrm{e}^{h(\log r)}$ if $h^{\prime \prime }(r)\to \infty $.
Publisher
Oxford University Press (OUP)
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