Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

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.

References [Enhancements On Off] (What's this?)

  • 1. F. Hartogs, Zur Theorie der analytischen Funktionen mehrerer unabhängiger Veränderlichen, Math. Ann. 62 (1906), 1-88. MR 1511365
  • 2. M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities. Cauchy's Equation and Jensen's Inequality, Uniwersytet Śläski - PWN, Warszawa - Kraków - Katowice, 1985 (Second edition, edited with a preface of Attila Gilányi, Birkhäuser Verlag, Basel, 2009). MR 788497 (86i:39008)
  • 3. J. Matkowski, Generalized convex functions and a solution of a problem of Zs. Páles, Publ. Math. Debrecen, 73/3-4 (2008), 421-460. MR 2466385 (2009i:26048)
  • 4. J. Matkowski, A mean-value theorem and its applications, J. Math. Anal. Appl. 373 (2011), 227-234. MR 2684472

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 26A24, 26E60, 39B22

Retrieve articles in all journals with MSC (2010): 26A24, 26E60, 39B22

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

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

American Mathematical Society