A Note on Schur-Convex Functions

Abstract

In this note it is proved that the integral arithmetic mean of a convex function is a Schur-convex function. Applications to Schur-convexity of logarithmic mean and gamma functions are given.

Schur-convexity and the Simpson formula

“…Namely, for a function f : [a, b] → R such that f (4) is continuous on [a, b], the following identity is valid:…”

“…For example, the following theorems were given in [4,5], respectively. Also, in [6] some related results were given.…”

Integral Inequalities Involving Strongly Convex Functions

“…These extensions and generalizations in the theory of inequalities have made valuable contributions in many areas of mathematics. Some new generalized concepts in this point of view are quasiconvex [1], strongly convex [2], approximately convex [3], logarithmically convex [4], midconvex functions [5], pseudoconvex [6], -convex [7], -convex [8], ℎ-convex [9], delta-convex [10], Schur convex [11][12][13][14][15], and others [16][17][18][19].…”

Schur-convexity and monotonicity of error of a general three-point quadrature formula

“…The first push in that direction was made in [6], where the following theorem was proven. Throughout this paper, by I we denote a nonempty open interval in R. A decade later, a related result regarding the midpoint and trapezoid formulas was obtained in [2].…”