A Combinatorial Refinement of Generalized Jessen's Inequality

Jessen's inequality is the functional form of Jensen's inequality for convex functions defined on an interval of real numbers. It is studied by many authors, but not always in a correct form. Jessen's inequality has a nice generalization based on totally normalised sublinear functionals. A brief overview of this topic is presented. The main result of this paper is a combinatorial improvement of the generalized Jessen's inequality. There are only a few refinements even for Jessen's inequality, our result gives a new type of refinement in a more general context. As an application we introduce and study new means.