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Claim. The answer to Problem 515 is yes for every entire function of finite order. Zhang Guanghou, Asymptotic values of entire and meromorphic functions, Sci. Sinica 20 (1977), 720--739 (the first zbMATH record linked above), proves that for every entire function $f$ of finite order that is not a polynomial there is a locally rectifiable path $C$ tending to infinity on which

$$\int_C\lvert f(z)\rvert^{-\lambda}\,\lvert\mathrm{d}z\rvert<\infty \quad\text{for every }\lambda>0$$

at once, one path serving every exponent. Lewis, Rossi and Weitsman cite this paper, their reference [12], for the case $u=\log\lvert f\rvert$ with $f$ of finite order of their Theorem 1, in the introduction recorded on [[problems/analysis/E0515/claims/1984_12_01_lewis_rossi_weitsman|Lewis, Rossi and Weitsman 1984]]; the statement follows that introduction and the site's account. The site's reference [Zh77] is the Kexue Tongbao item of the same title, author and year, Kexue Tongbao 22 (1977), 480, 486 as the site gives it, which the second zbMATH record linked above lists at p. 480, in Chinese.

Covers. The question for entire functions of finite order. Not covered: entire functions of infinite order, settled by the accepted full claim of Lewis, Rossi and Weitsman.

Depends on. Nothing in this wiki: the argument is the paper's own.

Acceptance. Refereed: the paper appeared in Scientia Sinica, a refereed journal. The site labels the problem PROVED and credits this paper with the finite-order case, while crediting Lewis, Rossi and Weitsman with the general case; the label settles the problem through their paper, not this one, so the page lists no reviewed evidence.

Dating. The page is dated by the publication year of the Scientia Sinica paper, 1977; the records give no issue month, and the month and day in the page name are placeholders.