Problem 1125

Let $f:\mathbb{R}\to \mathbb{R}$ be such that

2f(x) \leq f(x+h)+f(x+2h)

for every $x\in \mathbb{R}$ and $h>0$ . Must $f$ be monotonic?

Source: erdosproblems.com/1125

Analysis

Wiki page

Status

SolvedProved

An accepted solution exists. The statement is true.

PROVED (LEAN), on the site's label, which credits Laczkovich [La84] for the solution: every such function is nondecreasing, without any regularity assumption, by Laczkovich's published Theorem 1, recorded as an accepted claim; Kemperman's earlier proof of the measurable case [Ke69] is the accepted partial claim Kemperman 1969. The Lean part of the label refers to a public formalization of that proof, linked from the claim page and qualified under Public formal evidence below; this corpus has not built it.

Er81b: Erdős, P., My Scottish Book 'Problems'. The Scottish Book (1981), 27–35, problem cited by the site at p. 31. The page numbering is for the second edition of The Scottish Book.
Ke69: Kemperman, J. H. B., On the regularity of generalized convex functions. Transactions of the American Mathematical Society 135 (1969), 69–93.
La84: Laczkovich, M., On Kemperman's inequality $2f(x)\leq f(x+h)+f(x+2h)$ . Colloquium Mathematicum 49(1) (1984), 109–115. DOI 10.4064/cm-49-1-109-115. Source and full proof chain.

Problem 1125

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Proof claims

Discussion