Abstract/Summary

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 $e^{h(\log r)}$ if $h^{\prime\prime}(r)\to\infty$.