Theorem B (p. 498): the Lewis-Rossi-Weitsman path to infinity, as quoted

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Source. Theorem B, p. 498, of Jang-Mei Wu, Length of paths for subharmonic functions, J. London Math. Soc. (2) 32 (1985), 497--505, doi:10.1112/jlms/s2-32.3.497, the edition named on the source card. The paper quotes the theorem, without proof, from J. Lewis, J. Rossi and A. Weitsman, On the growth of subharmonic functions along paths, Ark. Mat. 22 (1984), 109--119. The paper says that article derives it as a consequence of Theorem A, the Lewis-Rossi-Weitsman form of Hall's lemma stated on p. 497.

Read depth. Claims checked: the quoted statement was read clause by clause on the page image of the print. The paper gives no proof, and the 1984 paper was not read for this page.

Statement

Let $u$ be subharmonic in $\mathbb C$, with $M(r)=\sup\{u(z):\lvert z\rvert=r\}$, such that $\lim_{r\to\infty}M(r)/\log r=\infty$. Then there is a path $\Gamma$ from $0$ to $\infty$ on which

$$\frac{u(z)}{\log\lvert z\rvert}\to\infty\quad\text{as }z\to\infty$$

and

$$\int_\Gamma e^{-\delta u(z)}\,\lvert dz\rvert<+\infty\quad \text{for each }\delta>0.$$

The path $\Gamma$ does not depend on $\delta$. The paper notes (p. 498) that weaker results for $u=\log\lvert f\rvert$, $f$ entire, were obtained by Huber and by Chang, and that its own Theorem 2 improves Theorem B when $u$ has positive lower order.

Bears on

Problem 514: the paper does not mention Erdős or the problem. Chojecki's note on the problem quotes this theorem from Wu's paper (as the note's Theorem 4) and applies it to $u=\max\{\log\lvert f\rvert,-1\}$ for transcendental entire $f$; that use is recorded on the claim page Chojecki 2026, and the theorem itself is credited to [[../wiki/problems/analysis/E0514/claims/1984_12_01_lewis_rossi_weitsman|Lewis, Rossi and Weitsman 1984]].