Invariance of the Cauchy mean-value expression with an application to the problem of representation of Cauchy means

Berrone, Lucio R.

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International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 18, Pages 2895-2912
doi:10.1155/IJMMS.2005.2895

Invariance of the Cauchy mean-value expression with an application to the problem of representation of Cauchy means

Lucio R. Berrone

Consejo Nacional de Investigaciones Científicas y Técnicas, Departamento de Matemática, Facultad de Ciencias Exactas, Ingenieria y Agrimensura, Universidad Nacional de Rosario, Avenue Pellegrini 250, Rosario 2000, Argentina

Received 30 August 2004; Revised 11 May 2005

Copyright © 2005 Hindawi Publishing Corporation. 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

The notion of invariance under transformations (changes of coordinates) of the Cauchy mean-value expression is introduced and then used in furnishing a suitable two-variable version of a result by L. Losonczi on equality of many-variable Cauchy means. An assessment of the methods used by Losonczi and Matkowski is made and an alternative way is proposed to solve the problem of representation of two-variable Cauchy means.