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Claim. Hayman's conjecture on the minimal growth holds: if a nonconstant entire $f$ is bounded outside a set of finite planar measure, that is, $|E(c)|<\infty$ for some $c$ with $E(c)=\{z:|f(z)|>c\}$, then $\int^\infty r\,dr/\log\log M(r,f)<\infty$. Hayman and Lingham's 2018 survey of Hayman's problems (library card, Update 2.40) states the conjecture with its sharpness as Camera's result and then credits Hansen and Gol'dberg with independent proofs. The paper is not held in the library; this page records the growth bound, which the paper's title and the survey's credit cover, and leaves the sharpness half unrecorded for Hansen, so the claim is the bound, not a determination of the minimal growth.

Covers. The growth bound of the first question, the necessary condition, as the survey credits it; not its sharpness, so the minimal growth rate is not determined by this claim alone. Camera's proof, with the sharpness statement, is on Camera's page; the second question's negative answer and a further independent proof of the bound are on Gol'dberg's page.

Source. L. J. Hansen, On the growth of entire functions bounded on large sets, Canad. J. Math. 29 (1977), no. 6, 1287--1291; the publisher's record is the linked DOI. The record dates the issue to December 1977, and the page is named by that date.

Acceptance. Refereed: the paper appeared in the Canadian Journal of Mathematics. Reviewed: Hayman and Lingham's survey credits Hansen with an independent proof of Hayman's conjecture. The site's commentary does not mention Hansen, so no curator credit is listed. Nothing here is independently reviewed by this project.

Depends on. Nothing on the wiki.