Jensen: a convexity framework for weighted means, including functions with corners

What is recorded here?

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. Averaging inputs and then applying a curved rule can give a different answer from applying the rule first and then averaging. A convex rule bends upward: its value at an average is no larger than the corresponding average of its values. Jensen’s paper builds a common framework for this fact and its many inequalities of means. It starts from midpoint comparisons, repeats them to obtain finite rational averages, and uses continuity to pass to arbitrary positive weights. An integral version follows under the paper’s bounded, integrable setting. The framework also handles corners such as an absolute-value function, so it does not require a derivative everywhere. Regularity still matters: midpoint convexity alone is weaker than real-weight convexity. Jensen proves continuity from a finite upper bound and discusses one-sided derivatives, composition and several variables. In a final addition he explicitly acknowledges Hölder’s earlier weighted-mean result under differentiability assumptions. The historical advance assessed here is the convexity framework and its regularity-sensitive extensions and unification, rather than first discovery of every inequality that now bears a familiar name.

Source-linked catalogue note. A detailed explanatory guide has not yet been prepared for this entry.

Scientific scope

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.

Read the original work

• 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 pages38-40

This English explanation is an editorial reading view. It does not replace the complete original work or certify its science.

People and systems

J. L. W. V. Jensen

Source-listed contributors to this scoped work. Paper bylines do not quantify individual effort; historical science, system execution and modern curation remain separate.