Jensen Functional, Quasi-Arithmetic Mean and Sharp Converses of Hölder’s Inequalities

Abstract

In this article, we give sharp two-sided bounds for the generalized Jensen functional Jn(f,g,h;p,x). Assuming convexity/concavity of the generating function h, we give exact bounds for the generalized quasi-arithmetic mean An(h;p,x). In particular, exact bounds are determined for the generalized power means in terms from the class of Stolarsky means. As a consequence, some sharp converses of the famous Hölder’s inequality are obtained.