New generalizations of Popoviciu-type inequalities via new Green's functions and Montgomery identity

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

doi:10.1186/s13660-017-1379-y

New generalizations of Popoviciu-type inequalities via new Green's functions and Montgomery identity

J Inequal Appl. 2017;2017(1):108. doi: 10.1186/s13660-017-1379-y. Epub 2017 May 10.

Authors

Nasir Mehmood¹, Ravi P Agarwal², Saad Ihsan Butt¹, Josip Pečarić³

Affiliations

¹ Department of Mathematics, COMSATS, Institute of Information Technology, Lahore, Pakistan.
² Department of Mathematics, Texas A&M University - Kingsville, 700 University Blvd., Kingsville, USA.
³ Faculty of Textile Technology, University of Zagreb, Zagreb, 10000 Croatia.

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.

Keywords: Abel-Gontscharoff interpolating polynomial; Grüss upper bounds; Montgomery identity; Ostrowski-type bounds; Popoviciu’s inequality; new Green’s functions.