Let $f=\sum_{n=0}^\infty a_nz^n$ be an entire function. Is it true that if
\[\lim_{r\to \infty} \frac{\max_n\lvert a_nr^n\rvert}{\max_{\lvert z\rvert=r}\lvert f(z)\rvert}\]
exists then it must be $0$?

Clunie (unpublished) proved this if $a_n\geq 0$ for all $n$. This was disproved in general by Clunie and Hayman

[ClHa64], who showed that the limit can take any value in $[0,1/2]$.

See also [513].