Some Remarks on Results Related to ∇-Convex Function

Abstract

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.

Keywords

• convex function
• ∇-convex function
• ∇-operator
• divided difference of a function
• finite difference of a function