TY - JOUR

T1 - Entropic means

AU - Ben-Tal, Aharon

AU - Charnes, Abraham

AU - Teboulle, Marc

N1 - Funding Information:
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. Carbon (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 (Mafh. 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),2 65-280i)s valid for entropic means. c 1989 Academic Press, lnc * This work was supported in part by the NSF Grant ECS-860-4354. ’ Part of this work was conducted while visiting the Center for Cybernetic Studies, The University of Texas, Austin. The author greatly appreciates the hospitality and support provided at this Institution.

PY - 1989/5/1

Y1 - 1989/5/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=38249023263&partnerID=8YFLogxK

U2 - 10.1016/0022-247X(89)90128-5

DO - 10.1016/0022-247X(89)90128-5

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AN - SCOPUS:38249023263

SN - 0022-247X

VL - 139

SP - 537

EP - 551

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 2

ER -