Erdős Problem #513

Let $f=\sum_{n=0}^\infty a_nz^n$ be a transcendental entire function. What is the greatest possible value of\[\liminf_{r\to \infty} \frac{\max_n\lvert a_nr^n\rvert}{\max_{\lvert z\rvert=r}\lvert f(z)\rvert}?\]

#513: [Er61,p.249]

analysis

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It is trivial that this value is in $[1/2,1)$. Kövári (unpublished) observed that it must be $>1/2$. Clunie and Hayman [ClHa64] showed that it is $\leq 2/\pi-c$ for some absolute constant $c>0$. Some other results on this quantity were established by Gray and Shah [GrSh63].

See also [227].

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This page was last edited 28 December 2025.

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