New Jensen-type inequalities

Abstract. We develop a new framework for the Jensen-type inequalities that allows us to deal with functions not necessarily convex and Borel measures not necessarily positive. It is well known the important role played by the classical inequality of Jensen in probability theory, economics, statistical physics, information theory etc. See [5] and [6]. In recent years, a number of authors have noticed the possibility to extend this inequality to the framework of functions that are mixed convex (in the sense of the existence of one inection point). See [1], [2] and [4]. In all these papers one assumes that both the function and the measure under consideration verify certain conditions of symmetry. However the inequality of Jensen is much more general as shows the following simple remark. Suppose that K is a convex subset of the Euclidean space RN carrying a Borel probability measure . Then every -integrable function f: K! R that admits a supporting hyperplane at the barycenter of ; (B) b = Z K xd(x); veri es the Jensen inequality (J) f(b) Z K f(x)d(x): Indeed, the existence of a supporting hyperplane at b is equivalent to the exis-tence of an a ¢ ne function h(x) = hx; vi+ c such that f(b) = h(b) and f(x) h(x) for all x 2 K