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

• Published in: Journal of inequalities and applications Vol. 2022; no. 1; pp. 1 - 15
• Main Authors: Bibi, Fazilat, Bibi, Rabia, Nosheen, Ammara, Pečarić, Josip
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 28-04-2022; Springer Nature B.V; SpringerOpen
• Subjects: Analysis; Applications of Mathematics; Diamond integrals; Diamonds; Functionals; Green's functions; Green’s function; Hermite polynomial; Hermite polynomials; Inequality; Interpolation; Jensen’s inequality; Mathematical analysis; Mathematics; Mathematics and Statistics; Time scales; Hermite polynomial; Jensen’s inequality; Green’s function; Time scales; 34N05; 28A25; Diamond integrals; 26D15
• ISSN: 1029-242X; 1025-5834; 1029-242X
• DOI: 10.1186/s13660-022-02785-1

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.