Some Remarks on Results Related to ∇-Convex Function

Asif Raza Khan (Department of Mathematics, University of Karachi, University Road, Karachi 75270, Pakistan)
Faraz Mehmood (Department of Mathematics, Dawood University of Engineering and Technology, New M.A. Jinnah Road, Karachi 74800, Pakistan)
Faisal Nawaz (Department of Mathematics, Dawood University of Engineering and Technology, New M.A. Jinnah Road, Karachi 74800, Pakistan)
Aamna Nazir (Department of Mathematics, University of Karachi, University Road, Karachi 75270, Pakistan)

Journal of Mathematical and Fundamental Sciences
Vol. 53 No. 1 (2021)

In the present article, we give new techniques for proving general identities of the Popoviciu type for discrete cases of sums for two dimensions using higher-order ∇-divided difference. Also, integral cases are deduced by different methods for differentiable functions of higher-order for two variables. These identities are a generalization of various previously established results. An application for the mean value theorem is also presented.