Popoviciu type characterization of positivity of sums and integrals for convex functions of higher order

Abstract

Abstract. Some very general identities of Abel and Popoviciu type for sumsUsing obtained identities, positivity of these expressions are characterized for convex functions of higher order. An application in terms of exponential convexity is given.Mathematics subject classification (2010): 26A51, 39B62, 26D15, 26D20, 26D99.

Some Remarks on Results Related to ∇-Convex Function

“…We have obtained the necessary and sufficient conditions by using the results of the previous theorem, where ( ) ≥ 0 holds ∀ ( + 1, + 1) − −convex function (see [1]). Now we recall a result from [10]. .…”

“…Now we state the main integral identity of this section using higher-order derivatives. We prove the following result in two different ways, first by twodimensional induction and then by Taylor expansion, recalling Theorem 4.2 from [10].…”

“…Let us recall some useful definitions and significant results from [1,8,10,11,16]. Throughout this article, is an interval in ℝ.…”

Weighted Montgomery's identities for higher order differentiable functions of two variables

“…For our next theorem, we give a lemma from [13] using our notations as follows. We give generalizations of Theorems 1.1, 1.2 and 1.3 respectively as follows:…”