This is Problem 2.40 in [Ha74] where it is attributed to Erdős. Hayman conjectured that\[\int_0^\infty \frac{r}{\log\log M(r)}\mathrm{d}r<\infty\]is true, and best possible, where $M(r)=\max_{\lvert z\rvert=r}\lvert f(z)\rvert$.

Hayman's strong conjecture was proved independently by Camera [Ca77] and Gol'dberg [Go79b].

The second question was answered in the negative by Gol'dberg [Go79b], who proved that if $T=\{ c>0 : \lvert E(c)\rvert <\infty\}$ then for any $m>0$ there exist entire functions $f$ such that $T=[m,\infty)$ or $T=(m,\infty)$. (It is clear that $T=\emptyset$ and $T=(0,\infty)$ are also possible.)

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This page was last edited 29 December 2025.