Majorization Inequalities via Peano's Representation of Hermite's Polynomial

Naveed Latif ,Nouman Siddique ,Josip Pečarić

doi:10.28924/2291-8639-16-2018-374

Abstract

The Peano's representation of Hermite polynomial and new Green functions are used to construct the identities related to the generalization of majorization type inequalities in discrete as well as continuous case. $\check{C}$eby$\check{s}$ev functional is used to find the bounds for new generalized identities and to develop the Gr$\ddot{u}$ss and Ostrowski type inequalities. Further more, we present exponential convexity together with Cauchy means for linear functionals associated with the obtained inequalities and give some applications.

Highlights

Introduction and PreliminariesNewton and Lagrange gave the classical methods for constructing Hermite interpolating polynomial
Lagrange gave the method for such function f (t) is defined at the distinct increasing points a1, a2, ..., an but Newton gave the method for such function f (t) is defined at the distinct points a1, a2, ..., an
We start with a brief overview of divided differences and n-convex functions and give some basic results from the majorization theory

Summary

Introduction

Introduction and PreliminariesNewton and Lagrange gave the classical methods for constructing Hermite interpolating polynomial. Let −∞ < α = a1 < a2 · · · < ar = β < ∞, (r ≥ 2) be the given points, w = (w1, ..., wm), x = (x1, ..., xm) and y = (y1, ..., ym) be m-tuples such that xl, yl ∈ [α, β], wl ∈ R (l = 1, ..., m) and Hij, Gc(c = 1, 2, 3, 4) be as defined in (1.10) and (2.2)-(2.5) respectively.

Results

Conclusion