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Mean-value theorems and some symmetric means

Matkowski J.

Demonstratio Mathematica

2013

Vol. 46, nr 3

461--474

Some variants of the Lagrange and Cauchy mean-value theorems lead to the conclusion that means, in general, are not symmetric. They are symmetric iff they coincide (respectively) with the Lagrange and Cauchy means. Under some regularity assumptions, we determine the form of all the relevant symmetric means.

Generalizations of Lagrange and Cauchy Mean-Value theorems

Matkowski J.

Demonstratio Mathematica

2010

Vol. 43, nr 4

765-774

Some generalizations of the Lagrange Mean-Value Theorem and Cauchy Mean-Value Theorem are proved and the extensions of the corresponding classes of means are presented.