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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 $\int^\infty r\,dr/\log\log M(r,f)<\infty$, and this is best possible: for every continuous positive nondecreasing $\phi$ with $\int^\infty r\,dr/\phi(r)<\infty$ there is an entire $f$ with $\log\log M(r,f)=O(\phi(r))$ and $|E(c)|<\infty$ for every $c>0$. The sharpness is stated here in the form of Part 2 of Gol'dberg's 1979 paper, whose closing note (p. 517) records that Camera's thesis proved the same two parts, the bound and its sharpness, as well as the analogue for functions subharmonic in $\mathbb R^m$. Hayman and Lingham's 2018 survey of Hayman's problems (library card, Update 2.40) attributes the conjecture, true and best possible, to Camera's thesis, but states the sharpness with the condition $\int^\infty r\,dr/\phi(r)=\infty$, a misprint: an entire $f$ with $\log\log M(r,f)<\phi(r)$ under that condition would have $\int^\infty r\,dr/\log\log M(r,f)=\infty$, contradicting the bound in the same sentence. The thesis is not held in the library; the claim is recorded as the survey, Gol'dberg's note and the site's commentary report it.

Covers. The first question, the minimal growth rate: Hayman's conjectured bound and its sharpness. The second question, whether finiteness of $|E(c)|$ passes to some $c'<c$, is not treated; its negative answer and an independent proof of the growth bound are on Gol'dberg's page.

Source. G. Camera, On the minimum rate of growth of certain classes of integral and subharmonic functions, PhD thesis, Imperial College, University of London (1977); the site's reference list prints the title with "classes on integral" [sic], and the survey's reference list gives the work as a 1977 University of London doctoral thesis without a title. The page is named by that year.

Acceptance. Reviewed: the site's curator, T. F. Bloom, credits Camera and Gol'dberg with independent proofs of Hayman's strong conjecture, and Hayman and Lingham's survey credits Camera with establishing it; Gol'dberg's paper records the same. A thesis is not a refereed publication, so no refereed evidence is listed. Nothing here is independently reviewed by this project.

Depends on. Nothing on the wiki.