Table of Contents
ISRN Mathematical Analysis
Volume 2011, Article ID 202409, 13 pages
http://dx.doi.org/10.5402/2011/202409
Research Article

Generalized Homogeneity of Means

1Computer Systems Department, Institute of Computer Science, University of Silesia, Będzińska 39, 41-200 Sosnowiec, Poland
2Institute of Mathematics, University of Silesia, Bankowa 14, 40-007 Katowice, Poland

Received 28 December 2010; Accepted 16 February 2011

Academic Editors: A. Levy, W. Sun, and J. Wu

Copyright © 2011 Małgorzata Pałys and Maciej Sablik. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We extend the classical notions of translativity and homogeneity of means to ๐น -homogeneity, that is, invariance with respect to an operation ๐น โˆถ ๐ผ ร— ๐ผ โ†’ ๐ผ . We find the shape of ๐น for the arithmetic weighted mean and then the general form of ๐น for quasi-linear means. Also, we are interested in characterizations of means. It turns out that no quasi-arithmetic mean can be characterized by ๐น -homogeneity with respect to a single operation ๐น , one needs to take two of such operations in order to characterize a mean.