On some inequalities for functions with nondecreasing increments of higher order

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-Pe ari ’s results, and a result for the integral mean of a function with nondecreasing increments of higher order.