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ASYMPTOTICS OF ENTIRE FUNCTIONS AND A PROBLEM OF HAYMAN
Quarterly Journal of Mathematics ( IF 0.7 ) Pub Date : 2020-04-03 , DOI: 10.1093/qmathj/haz061
Titus Hilberdink 1
Affiliation  

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 $|⁠.

中文翻译:

整函数的渐近性和海曼问题

在本文中,我们研究了半径为| $ r $ |的圆盘上的最大值的所有函数| $ e ^ {h(\ log r)} $ | 对于某些功能| $ h(\ cdot)$ |⁠。我们证明如果| $ h ^ {\ prime \ prime}(r)$ | 倾向于限制为| $ r \ to \ infty $ |⁠,从而解决了1966年的Hayman问题。另一方面,我们表明,在某些平缓的平滑条件下,整个函数可以像| $ \ textrm { e} ^ {h(\ log r)} $ | 如果| $ h ^ {\ prime \ prime}(r)\到\ infty $ |⁠
更新日期:2020-04-03
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