Recent research on Levinson’s inequality

In this paper we review the results on Levinson’s inequality and present some of its generalizations using several new approaches. We provide a probabilistic version 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. From the obtained inequality, we derive new families of exponentially convex functions and related results. We also give a monotonic refinement of the probabilistic version of Levinson’s inequality. Levinson’s type inequality of Hilbert space operators is discussed as well for unital fields of positive linear mappings and a large class of functions. Order among quasi- arithmetic means is considered under similar conditions.