OPEN

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

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 Kövári (unpublished) observed that it must be $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].