Generalized Identities and Inequalities of Čebyšev and Ky Fan Type for ∇−convex function

Faraz Mehmood\(^{a,*}\), Asif R. Khan\(^b\)

\(^a\)Department of Mathematics, Dawood University of Engineering and Technology, New M. A Jinnah Road, Karachi-74800, Pakistan.

\(^b\)Department of Mathematics, University of Karachi, Karachi, Pakistan.

Correspondence should be addressed to: faraz.mehmood@duet.edu.pk

Copyright © 2022 Faraz Mehmood , Asif R. Khan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Published: Received: : 5 June 2021; Accepted: 21 August 2022; Published Online: 31 August 2022.

Abstract

In the present article we establish three generalizations, first generalization is related to discrete Čebyšev identity for function of higher order ∇ divided difference with two independent variables and give its special case as a sequence of higher order ∇ divided difference. Moreover, we deduce results of discrete inequality of Čebyšev involving higher order ∇−convex function. The second and third generalizations are for integral Čebyšev and integral Ky Fan identities for function of higher order derivatives with two independent variables and discuss its inequalities using ∇−convex function. Generalized results give similar results of Pěcari´c’s article [23] and recapture some established results.

Keywords:

convex function, Čebyšev’s inequality, Ky Fan’s inequality.