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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Power means generated by some mean-value theorems


Author: Janusz Matkowski
Journal: Proc. Amer. Math. Soc. 139 (2011), 3601-3610
MSC (2010): Primary 26A24, 26E60; Secondary 39B22
Published electronically: March 9, 2011
MathSciNet review: 2813390
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Abstract | References | Similar Articles | Additional Information

Abstract: According to a new mean-value theorem, under the conditions of a function $ f$ ensuring the existence and uniqueness of Lagrange's mean, there exists a unique mean $ M$ such that

$\displaystyle \frac{f(x)-f(y)}{x-y}=M\left( f^{\prime}(x),f^{\prime}(y)\right). $

The main result says that, in this equality, $ M$ is a power mean if, and only if, $ M$ is either geometric, arithmetic or harmonic. A Cauchy relevant type result is also presented.


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Additional Information

Janusz Matkowski
Affiliation: Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Podgórna 50, PL-65246 Zielona Góra, Poland – and – Institute of Mathematics, Silesian University, Bankowa 14, PL-42007 Katowice, Poland
Email: J.Matkowski@wmie.uz.zgora.pl

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-10981-X
PII: S 0002-9939(2011)10981-X
Keywords: Mean-value theorem, mean, quasi-arithmetic mean, arithmetic mean, geometric mean, harmonic mean, Jensen functional equation, differential equation.
Received by editor(s): August 26, 2010
Published electronically: March 9, 2011
Communicated by: Edward C. Waymire
Article copyright: © Copyright 2011 American Mathematical Society