Extended Jensen’s functional for diamond integral via Green’s function and Hermite polynomial

Abstract

In this paper, with the help of Green’s function and Hermite interpolating polynomial, an extension of Jensen’s functional for n-convex functions 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 the further extensions of Jensen’s functional. This paper also includes discussion on bounds for Grüss inequality, Ostrowski inequality, and Čebyšev functional associated to the newly defined Jensen’s functional.

Generalizations of the Jensen functional involving diamond integrals via Abel–Gontscharoff interpolation

“…For example, when T = R the inequality (8) holds with Jensen's functional (2). Similarly, for T = Z the inequality (8) holds with Jensen's functional (3); for T = hZ, where h > 0, the inequality (8) holds with Jensen's functional (4); for T = q N 0 , where q > 1, the inequality (8) holds with Jensen's functional (5).…”

“…The number of research papers where Jensen's inequality is used is not countable. Several generalizations and refinements of Jensen's inequality for time-scales integrals can also be found in the literature (see [1,5,6,11,16,18]). For diamond integrals, Jensen's inequality is generalized in 2017, see [4].…”