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Statement

Notation (p. 22). Throughout Chapter 2, $f$ is entire and $M(r,f)=\max_{|z|=r}|f(z)|$.

Problem 2.41 (p. 38, quoted). "Suppose that $f(z)$ has finite order, and that $\Gamma$ is a rectifiable path on which $f(z)\to\infty$. Let $\ell(r)$ be the length of $\Gamma$ in $|z|<r$. Find such a path for which $\ell(r)$ grows as slowly as possible, and estimate $\ell(r)$ in terms of $M(r,f)$. If $f(z)$ has zero order, or more generally, finite order, can a path be found for which $\ell(r)=O(r)$ as $r\to\infty$? If $\log M(r,f)=O(\log^2r)$ as $r\to\infty$, but under no weaker growth condition, it is shown by Hayman [394] and Piranian [635] that we may choose a ray through the origin for $\Gamma$. If $f(z)$ has a finite asymptotic value $a$, the corresponding question may be asked for paths on which $f(z)\to a$."

The book's [394] is W. K. Hayman, Slowly growing integral and subharmonic functions, Comment. Math. Helv. 34 (1960), 75--84, and its [635] is G. Piranian, An entire function of restricted growth, Comment. Math. Helv. 33 (1959), 322--324. The book attributes the problem to P. Erdős, and Table 2 (p. 253) lists it among the problems of the 1974 symposium list.

Update 2.41 (p. 38). The update calls the problem a refined form of Problem 2.7 and says Gol'dberg and Eremenko (the book's [319]) solved it completely: for every function $\phi(r)$ tending to infinity there is an entire $f$ with $\ell(r)\ne O(r)$ for every asymptotic curve. The update does not restate how $\phi$ bounds the growth of $f$; Update 2.7 (p. 25) gives it as $T(r,f)/(\log r)^2\to\infty$ arbitrarily slowly. For the finite-value question the update adds that for every $\rho>1/2$ some entire function of order $\rho$ has a finite asymptotic value $a$ with $\ell(r)\ne O(r)$ for every asymptotic curve on which $f(z)\to a$; Update 2.7 credits Gol'dberg and Eremenko with such examples of order arbitrarily close to $\frac12$.

Source. W. K. Hayman and E. F. Lingham, Research Problems in Function Theory, arXiv:1809.07200v2 (21 September 2018), Chapter 2, p. 38. The edition read is identified on the source card.

Read depth. Claims checked: the notation, the problem, its update and the cited reference entries were read clause by clause on the printed pages. The book proves nothing; it poses and reports.

Proof pointer

None; a problem. The Gol'dberg–Eremenko theorems are on the Gol'dberg–Eremenko card.

Dependencies

Problem 2.7, of which the update calls this problem a refined form.

Bears on

• Problem 1115: the problem's statement follows the first paragraph of Problem 2.41, from Hayman's 1974 list, without the zero-order clause and with $\ell(r)\ll r$ for $O(r)$. Update 2.41 records the Gol'dberg–Eremenko negative answer to the linear-length question; it supplies no estimate of $\ell(r)$ in terms of $M(r,f)$.