Extended Jensen’s Functional for Diamond Integral via Hermite Polynomial

Abstract

In this paper, with the help of Hermite interpolating polynomial, extension of Jensen’s functional for $n$ -convex function is deduced from Jensen’s inequality involving diamond integrals. Special Hermite conditions, including Taylor two-point formula and Lagrange’s interpolation, are also deployed to find further extensions of Jensen’s functional. The paper also includes discussion on bounds for Grüss-type inequality, Ostrowski-type inequality, and $\breve{\text{C}}$ ebyšev functional associated with newly defined Jensen’s functional.