The extended mean values: Definition, Properties, Monotonicities, Comparison, Convexities, Generalizations, and Applications

Abstract

Mean values play important roles in the theory of inequalities, and even in the whole of mathematics, since many norms in mathematics are always means. Study of the extended mean values E(r, s; x, y) is not only interesting but important, both because most 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 expository article, we summarize the recent main results regarding the study of E(r, s; x, y) including its definition, basic properties, monotonicities, comparison, logarithmic convexities, Schur-covexities, generalizations of concepts of mean values, applications to quantum, to theory of special functions, to stablishment of Steffensen pairs, and to generalizarion of Hermite-Hadamard's inequality.