Jensen's inequality for nonconvex functions ∗

Abstract . Jensen’sinequalityisformulatedforconvexifiable(gen-erallynonconvex)functions. Keywords: Jensen’sinequality, convexifiable function, arithmeticmeantheorem AMSsubjectclassifications:26B25, 52A40Received June 23, 2004 Accepted November 2, 2004 1. Introduction Jensen’s inequality is 100 years old, e.g., [1, 2, 3] . It says that the value of a convexfunction at a point, which is a convex combination of finitely many points, is lessthan or equalto the convex combination of values of the function at these points.Using symbols: If : R n → R is convex then f pi =1 λ i x i ≤ pi =1 λ i f (x i )(1)for every set of p points x i ,i =1 ,...,p, in the Euclidean space R n and for all realscalars λ i ≥ 0, i =1 ,...,p , such that pi =1 λ i =1.In this note the inequality (1) is extended from convex to convexifiable func-tions, e.g., [4, 5]. These include all twice continuously differentiable functions, allonce continuously differentiable functions with Lipschitz derivative and all analyticfunctions. As a specialcase we obtain a new form of the arithmetic mean theorem.