SWhat changed
Jensen organized inequalities of weighted means around midpoint convexity, used continuity to extend rational averages to positive real weights and integrals, and derived regularity from upper boundedness.
Scientific scope
Jensen organized inequalities of weighted means around midpoint convexity, used continuity to extend rational averages to positive real weights and integrals, and derived regularity from upper boundedness. The inequality uses convexity and the stated nonnegative weights/normalization and domain assumptions. Equality and strictness depend on the function and input support. It is not valid for arbitrary signed weights or arbitrary functions, nor a guarantee that every optimization objective is convex. Earlier and later forms retain separate credit.
Evidence
Source-linked, scoped account of the original work. The published scope states the scientific assumptions and remaining verification limits. WRS has not independently certified the full proof, experimental result or system benchmark.
Contributors
- J. L. W. V. Jensen · SCIENCE: Original source-listed scientific author of this scoped work; byline does not determine effort fractions.
- J. L. W. V. Jensen, Sur les fonctions convexes et les inégalités entre les valeurs moyennes — full French original
- O. Hölder, Ueber einen Mittelwerthssatz (1889) — original-text transcription, selected pages 38-40
Provisional WRS assessment, edition 2026-10-16.1. Independent expert calibration is pending. This grade does not certify correctness or assign ownership.
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