Abstract. We study necessary and sufficient conditions on linear operators A and B for inequality A f B f to hold for every function f that is n -convex at a point.Mathematics subject classification (2010): 26D99, 26D15.

Abstract. Mercer [5] gave a generalization of Levinson's inequality that replaces the assumption of symmetry of the two sequences with a weaker assumptions of equality of variances. Witkowski [10] further loosened this assumption and extended the result to the class of 3-convex functions.We generalize these results to a newly defined, larger class of functions. We also prove the converse in case the function is continuous. In particular, we show that if Levinson's inequality holds under Mercer's assumptions, then the function is 3-convex.Mathematics subject classification (2010): 26D15.

Abstract. We give a generalization of Steffensen's inequality by extending the results of Pečarić [4] and Rabier [5]. We make use of the n -order Taylor expansion of a composition of functions and Faà di Bruno's formula for higher order derivatives of the composition.Mathematics subject classification (2010): 26D10, 26D15.

Abstract. In this paper, we will give general Hardy and reversed Hardy type inequalities for a generalized class of monotone functions. Moreover we will give n -exponential convexity, exponential convexity and related results for some functionals obtained from the differences of these inequalities. At the end we will give mean value theorems and Cauchy means for these functionals.Mathematics subject classification (2010): 26D15, 26A48.

Abstract. We give a probabilistic version of Levinson's inequality under Mercer's assumption of equal variances for the family of 3-convex functions at a point. We also show that this is the largest family of continuous functions for which the inequality holds. New families of exponentially convex functions and related results are derived from the obtained inequality.