Entropic means

Aharon Ben-Tal*, Abraham Charnes, Marc Teboulle

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


We show how to generate means as optimal solutions of a minimization problem whose objective function is an entropy-like functional. Accordingly, the resulting mean is called entropic mean. It is shown that the entropic mean satisfies the basic properties of a weighted homogeneous mean (in the sense of J. L. Brenner and B. C. Carlson (J. Math. Anal. Appl., 123 (1987), 265-280)) and that all the classical means as well as many others are special cases of entropic means. Comparison theorems are then proved and used to derive inequalities between various means. Generalizations to entropic means of random variables are considered. An extremal principle for generating the Hardy-Littlewood-Polya generalized mean is also derived. Finally, it is shown that an asymptotic formula originally discovered by L. Hoehn and I. Niven (Math. Mag., 58 (1958), 151-156) for power means, and later extended by J. L. Brenner and B. C. Carlson (J. Math. Anal. Appl., 123 (1987), 265-280) is valid for entropic means.

Original languageEnglish
Pages (from-to)537-551
Number of pages15
JournalJournal of Mathematical Analysis and Applications
Issue number2
StatePublished - 1 May 1989
Externally publishedYes


FundersFunder number
National Science FoundationECS-860-4354


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