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Statement

Notation (p. 22). Throughout Chapter 2, $f$ is entire, and $a$ is an asymptotic value of $f$ if $f(z)\to a$ as $z\to\infty$ along a path $\Gamma$, an asymptotic path; by Iversen's theorem $\infty$ is an asymptotic value of every entire function.

Problem 2.7 (p. 25, quoted). "If $f(z)$ [sic] of finite order, can anything be asserted about the length of $\Gamma_\infty$, which is the path on which $f(z)$ tends to $\infty$, or the part of it in $|z|\leq r$?"

The problem carries no attribution line, and Table 2 (p. 253) lists it among the problems of the 1967 edition.

Update 2.7 (p. 25). The update measures $\ell(r)$ as the length of the arc of $\Gamma_\infty$ up to its first intersection with $|z|=r$. It reports:

• Hayman (the book's [394]): if $T(r,f)=O((\log r)^2)$ as $r\to\infty$, then $\Gamma_\infty$ can be taken to be a straight line.
• Eremenko and Gol'dberg (the book's [319]): examples with $T(r,f)/(\log r)^2\to\infty$ arbitrarily slowly for which $\ell(r)=O(r)$ fails; an independent proof by Toppila (the book's [756]: 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).
• Chang Kuan Heo (the book's [152]: K. H. Chang, Asymptotic values of entire and meromorphic functions, Sci. Sinica 20 (1977), 720--739): if $f$ has finite order $\rho$, then for any $\varepsilon>0$ one can always have $\ell(r)=O(r^{1+\frac12\rho+\varepsilon})$.
• For a finite asymptotic value $a$ and a path $\Gamma_a$, Gol'dberg and Eremenko [319]: examples of order arbitrarily close to $\frac12$ with $\ell(r)\ne O(r)$.

It refers further to Update 2.10 and to Lewis, Rossi and Weitsman (the book's [518]). The book prints [319] as Mat. Sb. (N.S.) 79, 109 (151) (No. 4), 555--581, 1982; the paper appeared in Mat. Sb. 109 (151) (1979), as the Gol'dberg–Eremenko card records. Update 2.41 (p. 38) states the finite-value examples for every order $\rho>1/2$; see Problem 2.41.

Source. W. K. Hayman and E. F. Lingham, Research Problems in Function Theory, arXiv:1809.07200v2 (21 September 2018), Chapter 2, p. 25. 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.

Dependencies

None.

Bears on

• Problem 1115: Problem 2.7 is the earlier form of the question; Update 2.41 calls Problem 2.41, the source of #1115's wording, "a refined form of Problem 2.7". The problem page cites Update 2.7 for Toppila's independent proof and for Chang's bound, which is stated for the length up to the first intersection with $|z|=r$, not for the whole length inside the disc.