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The Extended Mean Values: Definition, Properties, Monotonicities, Comparison, Convexities, Generalizations, and Applications

[error in script] (2001) The Extended Mean Values: Definition, Properties, Monotonicities, Comparison, Convexities, Generalizations, and Applications. RGMIA research report collection, 5 (1).

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Abstract

The extended mean values E(r, s; x, y) play an important role in theory of mean values and theory of inequalities, and even in the whole mathematics, since many norms in mathematics are always means. Its study is not only interesting but important, both because most of the two-variable mean values are special cases of E(r, s; x, y), and because it is challenging to study a function whose formulation is so indeterminate. In this expositive article, we summarize the recent main results about study of E(r, s; x, y), including definition, basic properties, monotonicities, comparison, logarithmic convexities, Schur-convexities, generalizations of concepts of mean values, applications to quantum, to theory of special functions, to establishment of Steffensen pairs, and to generalization of Hermite-Hadamard’s inequality.

Item Type: Article
Uncontrolled Keywords: Extended mean values, generalized weighted mean values, generalized abstracted mean values, logarithmic convex, Schur-convex, monotonicity, comparison, definition, recurrence formula, integral expression, gamma function, incomplete gamma function, Steffensen pairs, Hermite-Hadamard inequality, absolutely (completely, regularly) monotonic (convex) function, arithmetic mean of function, quantum, Bernoulli’s numbers, Bernoulli’s polynomials
Subjects: FOR Classification > 0102 Applied Mathematics
FOR Classification > 0103 Numerical and Computational Mathematics
Collections > Research Group in Mathematical Inequalities and Applications (RGMIA)
Depositing User: Research Group in Mathematical Inequalities and Applications
Date Deposited: 30 Aug 2012 03:52
Last Modified: 23 May 2013 16:51
URI: http://vuir.vu.edu.au/id/eprint/17679
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