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Claim. The answer to Problem 516 is yes for every entire function $f(z)=\sum_{k\ge1}a_kz^{n_k}$ with $\sum_k1/(n_{k+1}-n_k)<\infty$. P. Erdős and A. J. Macintyre, Integral functions with gap power series, Proc. Edinburgh Math. Soc. (2) 10 (1954), no. 2, 62--70, doi:10.1017/S0013091500021416, prove in their Theorem 1 that for every entire function $f$ with strictly increasing exponents $n_k$ and

$$\sum_{k}\frac1{n_{k+1}-n_k}<\infty$$

one has $\limsup_{r\to\infty}m(r)/M(r)=\limsup_{r\to\infty}\mu(r)/M(r)=1$, where $M(r)$, $m(r)$ and $\mu(r)$ are the maximum modulus, the minimum modulus and the maximum term on $\lvert z\rvert=r$; the theorem has no order hypothesis. Along a sequence of radii with $m(r)/M(r)\to1$ one has $\log m(r)/\log M(r)\to1$, since $M(r)\to\infty$, and $m(r)\le M(r)$ gives the reverse bound, so $\limsup\log m(r)/\log M(r)=1$, the question's statement; and the convergence of the gap sum forces the gaps $n_{k+1}-n_k$ to tend to infinity, hence $n_k/k\to\infty$, so every finite-order $f$ with a convergent gap sum belongs to the question's class. The paper presents Theorem 1 as a sharpening of a remark in the last sentence of Pólya's 1929 paper, that $\limsup m(r)/M(r)=1$ holds when $\liminf_{k\to\infty}\log(n_{k+1}-n_k)/\log n_k>1/2$, and notes that Pólya's condition implies the convergent gap sum. Its Theorem 2 shows the condition sharp: whenever the gap sum diverges there is an entire function with those exponents and $\limsup\mu(r)/M(r)\le1/2$ and $\limsup m(r)/M(r)\le1/2$. The statement follows the paper's print (pp. 62--63); the proof is not reconstructed in this repository.

Covers. Entire functions of finite order with $\sum1/(n_{k+1}-n_k)<\infty$, a subclass of the question's class, with the stronger conclusion $\limsup m(r)/M(r)=1$. Not covered: finite-order functions with $n_k/k\to\infty$ and a divergent gap sum, settled by the accepted full claim of Fuchs.

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

Acceptance. Refereed: the paper appeared in the Proceedings of the Edinburgh Mathematical Society, a refereed journal. The site labels the problem PROVED (LEAN) and credits Fuchs with the solution, recording this paper's gap condition as an earlier result; the label settles the problem through Fuchs's paper, not this one, so the page lists no reviewed evidence.

Dating. The page is dated by the issue month in the publisher's record (Crossref), December 1954; the day is a placeholder.