Functions with Non-decreasing Increments and Popoviciu Type Identities and Inequalities for Sums and Integrals

Abstract

In past few decades, the topic of functions with non-decreasing increments has gained popularity in several branches of Mathematics and this topic got attention of many mathematicians due to its importance. We would like to establish the relationship(s) among functions with non-decreasing increments and arithmetic integral mean, Wright convex functions, convex functions, \(\nabla\)-convex functions, Jensen m-convex functions, m-convex functions, m-\(\nabla\)-convex functions, k-monotonic functions, absolutely monotonic functions, completely monotonic functions, Laplace Transform and exponentially convex functions, by using the finite difference operator as different cases of \(\Delta_{h}^{m}f\). We also consider function with nondecreasing increments of third order and obtain the generalizations of the Levinson’s-type inequality and Jensen-Mercer’s-type inequality by using Jensen-Boas inequality. We will deduce some general identities of Popoviciu type for discrete case for sums for function and sequence in two dimensions using higher order \(\nabla\) divided difference, positivity of these expressions are characterized for higher order \(\nabla\)-convex functions. We will also obtain some general identities of Popoviciu type for integral \(\displaystyle \iint P(y_{t}z)f(y_{t}z)dy~dz\) of higher order differentiable function and positivity of these expressions are characterized for higher order \(\nabla\)-convex and completely monotonic functions. We would discuss some applications in terms of generalized Cauchy-type means and exponential convexity as well. We would get the generalization of discrete identity and inequality of \(\check{C}\)eby\(\check{s}\)ev-type and discuss generalization of integral identities and inequalities of \(\check{C}\)eby\(\check{s}\)ev-type and Ky Fan-type for higher order \(\nabla\)-convex functions with two variables. Moreover, we will obtain double weighted integrals Montgomery’s identities with two variables and using obtained identities we deduce generalization of Ostrowski-type and Gruss-type inequalities for higher order differentiable functions.