Problem 1118

Let $f(z)$ be a non-constant entire function such that, for some $c$ , the set $E(c)=\{ z: \lvert f(z)\rvert >c\}$ has finite measure.

What is the minimum growth rate of $f(z)$ ?

If $E(c)$ has finite measure then must there exist $c'<c$ such that $E(c')$ has finite measure?

Source: erdosproblems.com/1118

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An accepted solution exists. Settled in another form, for example when its parts resolve differently or the question is open-ended.

Solved on the site. Gol'dberg's 1979 paper answers both questions: the minimal growth is Hayman's conjectured bound, $\int^\infty r\,dr/\log\log M(r)<\infty$ , proved and shown best possible, and the second question has the answer no (Gol'dberg's claim page (Goldberg, 1979)); Camera's 1977 thesis is credited with an independent proof of Hayman's conjecture, the bound and its sharpness (Camera's claim page (1977)), and Hayman and Lingham's survey of Hayman's problems (library card, Update 2.40) credits Hansen's 1977 paper with another, which the site does not mention (Hansen's claim page (1977)). The standing is solved, answered.

Problem 1118

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