Problem 227

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claims/: The 1 claim page of Problem 227, one per claimant's result; the problem's standing derives from them.

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Statement. Let $f=\sum_{n=0}^\infty a_nz^n$ be an entire function which is not a polynomial. 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 $0$?

Status. Disproved. The site credits Clunie and Hayman [ClHa64], who produce transcendental entire functions for which the ratio tends to any prescribed value in $[0,1/2]$; the claim page [[problems/analysis/E0227/claims/1964_12_01_clunie_hayman|Clunie and Hayman 1964]] records it, accepted on its refereed publication and the site's credit.

Source. erdosproblems.com/227, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #227, https://www.erdosproblems.com/227.

References.

• [ClHa64] Clunie, J. and Hayman, W. K., The maximum term of a power series. J. Analyse Math. (1964), 143-186.

Formalization. None recorded.

Current assessment

The site's formulation of 2026-09-04 asks whether the ratio of the maximum term $\max_n|a_n|r^n$ to the maximum modulus $M(r)$ of a transcendental entire function, when it has a limit as $r\to\infty$, must have limit $0$. Clunie and Hayman's 1964 paper answers no: the limit can be any value in $[0,1/2]$. The site also records that Clunie, in unpublished work, proved the answer yes when every coefficient $a_n$ is nonnegative; that case has no dated manuscript and no page here, and it does not affect the disproof of the general question. The related Problem 513 asks for the largest possible limit inferior of the same ratio. The standing rests on the refereed paper and the site's credit, as the claim page records. The paper is not held in the library, so no proof is compiled or reviewed here. No status search beyond the site is recorded.