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

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. Fritz Hartogs, Zur Theorie der analytischen Funktionen mehrerer unabhängiger Veränderlichen, insbesondere über die Darstellung derselben durch Reihen, welche nach Potenzen einer Veränderlichen fortschreiten, Math. Ann. 62 (1906), no. 1, 1–88 (German). MR 1511365,
  • 2. Marek Kuczma, An introduction to the theory of functional equations and inequalities, Prace Naukowe Uniwersytetu Śląskiego w Katowicach [Scientific Publications of the University of Silesia], vol. 489, Uniwersytet Śląski, Katowice; Państwowe Wydawnictwo Naukowe (PWN), Warsaw, 1985. Cauchy’s equation and Jensen’s inequality; With a Polish summary. MR 788497
  • 3. Janusz Matkowski, Generalized convex functions and a solution of a problem of Zs. Páles, Publ. Math. Debrecen 73 (2008), no. 3-4, 421–460. MR 2466385
  • 4. Janusz Matkowski, A mean-value theorem and its applications, J. Math. Anal. Appl. 373 (2011), no. 1, 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