New generalizations of Popoviciu-type inequalities via new Green\u2019s functions and Montgomery identity

Nasir Mehmood,Ravi P Agarwal,Josip Pečarić,Saad Ihsan Butt

doi:10.1186/s13660-017-1379-y

Abstract

The inequality of Popoviciu, which was improved by Vasić and Stanković (Math. Balk. 6:281-288, 1976), is generalized by using new identities involving new Green’s functions. New generalizations of an improved Popoviciu inequality are obtained by using generalized Montgomery identity along with new Green’s functions. As an application, we formulate the monotonicity of linear functionals constructed from the generalized identities, utilizing the recent theory of inequalities for n-convex functions at a point. New upper bounds of Grüss and Ostrowski type are also computed.

Highlights

Higher order convexity was introduced by Popoviciu, who defined it under the context of divided differences of a function
Inequalities of higher order convex functions are very important and many physicists used them while dealing with higher dimensions
It is interesting to note that results for convex functions may not be true for convex functions of higher order

Summary

Introduction

Higher order convexity was introduced by Popoviciu, who defined it under the context of divided differences of a function (see Ch. , [ ]). Popoviciu in their book [ ] in to commemorate years to Popoviciu’s inequality They generalized Popoviciu’s inequality for higher order convex functions and gave its applications. In [ ], a new class of n-convex functions at a point was introduced by Pečarić, Praljak and Witkowski. They developed a remarkable theory to investigate linear operator inequalities with the help of the functions, which are n-convex at a point. This theory leads to many interesting and fascinating results with a lot of applications in operator theory and statistics. We present our main results by introducing some new types of Green’s functions

Main results

Conclusions