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Claim. The first question of [[problems/analysis/E0514/_index|Problem 514]] has the answer yes. John Lewis, John Rossi and Allen Weitsman, On the growth of subharmonic functions along paths, Ark. Mat. 22 (1984), no. 1--2, 109--119, doi:10.1007/BF02384375, prove in their Theorem 1 that for every function $u$ subharmonic in the plane with $M(r,u)/\log r\to\infty$, where $M(r,u)=\max_{\lvert z\rvert=r}u(z)$, there is a path $\Gamma$ tending to infinity on which $u(z)/\log\lvert z\rvert \to\infty$ and $\int_\Gamma e^{-\lambda u}\,\lvert\mathrm{d}z\rvert<\infty$ for every $\lambda>0$. For a transcendental entire function $f$ the function $u=\log\lvert f\rvert$ is subharmonic with $\log M(r,f)/\log r\to\infty$, so there is a path $\Gamma$ to infinity with

$$\frac{\log\lvert f(z)\rvert}{\log\lvert z\rvert}\to\infty \quad(z\to\infty\text{ along }\Gamma), \qquad\text{hence}\qquad \Bigl\lvert\frac{f(z)}{z^n}\Bigr\rvert\to\infty \text{ for every }n,$$

one path serving every $n$. The paper presents Theorem 1 as a generalization of Huber's Theorem A, one path for each exponent, and of Talpur's Theorem B (M. N. M. Talpur, On the growth of subharmonic functions on asymptotic paths, Proc. London Math. Soc. (3) 32 (1976), no. 2, 193--198), which already gives, for every $u$ subharmonic in the plane with $M(r,u)/\log r\to\infty$, a path to infinity on which $u(z)/\log\lvert z\rvert\to\infty$; so the first question is already answered by Talpur's theorem with $u=\log\lvert f\rvert$, and this paper's addition is the integral condition on the same path. The same theorem settles Problem 515, and its claim page [[problems/analysis/E0515/claims/1984_12_01_lewis_rossi_weitsman|Lewis, Rossi and Weitsman 1984]] records the statement of Theorem 1 and the paper's introduction; the deduction above is the case $u=\log\lvert f\rvert$ of the path-growth clause. Wu's 1985 paper restates the theorem as its Theorem B, the form in which [[problems/analysis/E0514/claims/2026_04_20_chojecki|Chojecki's note]] applies it and adds the length estimate the second question asks for.

Covers. The first question (the part path), answered yes. Not covered: the length of the path in terms of $M(r)$ (the paper states no length bound, which is deduced from the integral in Chojecki's note) and the growth of $f$ along the path in terms of $M(r)$.

Depends on. Nothing in this wiki: the argument is the paper's own, and the specialization to $u=\log\lvert f\rvert$ is immediate.

Acceptance. Refereed: the paper appeared in Arkiv för Matematik, a refereed journal. The site labels the problem OPEN and credits Boas, unpublished, with the first question; it does not credit this paper, so the page lists no reviewed evidence.

Dating. The page is dated by the issue month in the publisher's record (Crossref), December 1984; the day is a placeholder.