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 f by arithmetic means of f 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.