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

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