Problem 1118

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claims/: The 3 claim pages of Problem 1118, one per claimant's result; the problem's standing derives from them.

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Statement. 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?

Status. 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); Camera's 1977 thesis is credited with an independent proof of Hayman's conjecture, the bound and its sharpness (Camera's claim page), 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). The standing is solved, answered.

Source. erdosproblems.com/1118, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1118, https://www.erdosproblems.com/1118.

References.

• [Ca77] G. Camera, On the minimum rate of growth of certain classes on integral and subharmonic functions, PhD Thesis. Imperial College, University of London (1977).
• [Go79b] Gol'dberg, A. A., Sets on which the modulus of an entire function has a lower bound. Sibirsk. Mat. Zh. 20 (1979), no. 3, 512–518, 691.
• [Ha74] Hayman, W. K., Research problems in function theory: new problems. (1974), 155-180.

Formalization. None recorded.

Progress

Not yet compiled.

Known Results

Not yet compiled.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• goldberg_1979_sets_which_modulus_entire_function_has
• goldberg_1979_sets_which_modulus_entire_function_has / section_1
• goldberg_1979_sets_which_modulus_entire_function_has / section_2
• goldberg_1979_sets_which_modulus_entire_function_has / section_3
• hayman_lingham_2018_research_problems_function_theory
• hayman_lingham_2018_research_problems_function_theory / problem_2_40