Further refinements of the Jensen inequalities based upon samples with repetitions

In this paper, we give a new refinement and extension of the well-known Jensen inequality for any mid-convex function ff by arithmetic means of ff evaluated at weighted arithmetic means of samples with repetitions. Our refinement provides the perfected and corrected version of the results given earlier by Pečarić and Svrtan (1998) [5]. As generalizations, we also obtain two other refinements of Jensen’s inequality.