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Claim. Let $B$ be the supremum, over transcendental entire functions $f$, of $\liminf_{r\to\infty}\mu(r,f)/M(r,f)$, where $\mu(r,f)$ is the maximum term of the power series of $f$ at radius $r$ and $M(r,f)$ its maximum modulus there; $B$ is the value Problem 513 asks for. J. Clunie and W. K. Hayman, The maximum term of a power series, J. Analyse Math. 12 (1964), 143--186, doi:10.1007/BF02807433, prove

$$\frac47<B\le\frac2\pi-c$$

for an absolute constant $c>0$: a transcendental entire function whose limit inferior exceeds $4/7\approx0.57143$, and an upper bound for every transcendental entire function that improves the bound $2/\pi$ obtained from an argument of Clunie reported by Gray and Shah in 1963. The constant $c$ is left unspecified in the site's account of the paper, which the statement above follows.

Covers. The two bounds only: $B>4/7$ and $B\le2/\pi-c$ for some absolute $c>0$. The value of $B$ is not determined, and the pending claim pages of the folder assert larger lower bounds.

Depends on. Nothing in this wiki: the argument is the paper's own.

Acceptance. Refereed: the paper appeared in the Journal d'Analyse Mathématique, a refereed journal. The site's commentary credits the two bounds to the paper, but the site labels the problem OPEN, so that credit is not acceptance of the problem and the page lists no reviewed evidence. Later lower bounds that supersede $4/7$ are [[problems/analysis/E0513/claims/2026_02_12_he_tang|He and Tang's certified lower bound]], [[problems/analysis/E0513/claims/2026_02_27_sothanaphan|Sothanaphan's certified parameter improvement]] and [[problems/analysis/E0513/claims/2026_08_02_lystad|Lystad's certified lower bound]], all pending; the upper bound stands.

Dating. The page is dated by the issue month in the publisher's record (Crossref), December 1964; the day is a placeholder.