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Claim. The answer to the question of Problem 1115 is no: an entire function $f$ of finite order, even of order zero, need not have a locally rectifiable path on which $f\to\infty$ whose length $\ell(r)$ inside $|z|<r$ is $O(r)$. S. Toppila, On the length of asymptotic paths of entire functions of order zero, Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980), 13--15, poses the question as Erdős's, citing Problem 2.41 of Hayman's 1974 collection, and recalls that under the growth condition $\log M(r,f)=O((\log r)^2)$ Hayman's 1960 theorem gives a ray through the origin as such a path (Hayman's claim page). Its Theorem states that for every increasing function $\varphi(r)\to\infty$ there is an entire function $f$ with

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

such that every locally rectifiable curve $P$ on which $f(z)\to\infty$ has $\limsup_{r\to\infty}\ell(r)/r=\infty$, where $\ell(r)$ is the length of $P$ in $|z|<r$. With $\varphi(r)=\log r$ the function has order zero, so the definite question has a negative answer already at order zero, and the note presents Hayman's condition as the best possible growth condition for a linear-length path. The two-page proof builds the function as an infinite product of polynomials chosen so that $|f|\le1$ on a sequence of spiral paths circling the origin, which every path on which $f\to\infty$ must eventually avoid, forcing $\ell(e r_n)\ge2\pi nr_n$ for large $n$. The note's closing remark records that the same result had been proved by Gol'dberg and Eremenko in their 1979 paper, which has its own claim page.

Depends on. Nothing in this wiki; the construction is self-contained.

Acceptance. Refereed: Annales Academiae Scientiarum Fennicae, Series A I Mathematica, volume 5 (1980), pp. 13--15; the journal's record dates the article 1 February 1980, the date this page carries. Hayman and Lingham's 2018 survey of Hayman's problems (Update 2.7) records the note as an independent proof of the Gol'dberg--Eremenko result. The site's curator credits Gol'dberg and Eremenko alone and does not mention the note, so no reviewed evidence is listed. No proof has been reproduced or reviewed in this wiki. The problem also asks for a path of slowest growth and an estimate of $\ell(r)$ in terms of $M(r)$; the note supplies no such estimate, and none is claimed here.