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Claim. For every nonconstant entire function $f$ with

$$\log M(r,f)=O\bigl((\log r)^2\bigr),$$

$f(z)\to\infty$ along the rays of almost every argument, so a ray through the origin is a path to infinity with $\ell(r)=r$ inside $|z|<r$, and the question of Problem 1115 has the answer yes for that class. The result is from W. K. Hayman, Slowly growing integral and subharmonic functions, Comment. Math. Helv. 34 (1960), 75--84; it is stated in this form on p. 509 of the English translation of Gol'dberg and Eremenko's 1979 paper, and the site's commentary credits it with a path on which $f\to\infty$ and $\ell(r)=r$ whenever $\log M(r)\ll(\log r)^2$. Toppila's 1980 note records the same consequence, a ray through the origin under this condition.

Covers. Entire functions with $\log M(r,f)=O((\log r)^2)$, a class of functions of order zero: for them a linear-length path exists, and a ray serves. Nothing outside that class. Gol'dberg and Eremenko's Theorem 1 (their claim page) and Toppila's Theorem (Toppila's claim page) show that no unbounded multiplicative relaxation of this growth condition always yields a linear-length path, not that every function outside the class fails; the problem's request for an estimate of $\ell(r)$ in terms of $M(r)$ for general finite order is not covered.

Depends on. Nothing in this wiki.

Acceptance. Refereed: Commentarii Mathematici Helvetici (the publisher's record: volume 34, issue 1, pp. 75--84, issued December 1960; the day is the issue's nominal first day, used for this page's date). The site's curator, Thomas Bloom, credits this theorem in the problem's commentary, but the site's label settles the problem by Gol'dberg and Eremenko's disproof, so that commentary is not listed as reviewed evidence. The original proof is not reproduced on the library's result pages or in this wiki. This claim is partial, so the problem's standing derives from the full claims.