We investigate a class of functions with nondecreasing increments of higher order. A generalization of Brunk’s theorem is proved for that class of functions. Also, we consider functions with nondecreasing increments of order three, we obtain the Levinson-type inequality, a generalization of Burkill-Mirsky-Pečarić’s results, and a result for the integral mean of a function with nondecreasing increments of higher order.

In this article, we give more generalized results than in Anwar et al. (2010) and Latif and Pečarić (2010) in new direction by using second-order divided difference. We investigate the exponential convexity and logarithmic convexity for majorization type results by using class of continuous functions in linear functionals. We also construct positive semi-definite matrices for majorization type results. We will vary on choice of a family of functions in order to construct different examples of exponentially convex functions and construct some means. We also prove the monotonic property.

Extension of Montgomery's identity is used in derivation of Popoviciu-type inequalities containing sums , where f is an n-convex function. Integral analogues and some related results for n-convex functions at a point are also given, as well as Ostrowski-type bounds for the integral remainders of identities associated with the obtained inequalities.