Abstract
Considering a weighted relation of majorization, Sherman obtained a useful generalization of the classical majorization inequality. The aim of this paper is to extend Sherman’s inequality to convex functions of higher order. An upper bound for Sherman’s inequality, as well as for generalized Sherman’s inequality, is given with some applications. As easy consequences, some new bounds for Jensen’s inequality can be derived.
Highlights
1 Introduction and preliminaries For any given partial ordering of a set χ, real-valued functions φ defined on χ, which satisfy φ(x) ≤ φ(y) whenever x y, are variously referred to as ‘monotonic’, ‘isotonic’, or ‘order-preserving’
We consider a partial ordering of majorization
We present an upper bound for generalized Sherman’s inequality which is an Ostrowski type inequality
Summary
For any given partial ordering of a set χ , real-valued functions φ defined on χ , which satisfy φ(x) ≤ φ(y) whenever x y, are variously referred to as ‘monotonic’, ‘isotonic’, or ‘order-preserving’. We consider a partial ordering of majorization. X[i] and y[i] denote the elements of x and y sorted in decreasing order. Majorization on vectors determines the degree of similarity between the vector elements. ). For the concept of majorization, the order-preserving functions were first systematically studied by Schur In his honor, such functions are said to be ‘convex in the sense of Schur’, ‘Schur convex’, or ‘S-convex.’
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