Generalizations of Some Classical Inequalities and Their Applications

Abstract

The main aim of this talk is to review, complement and generaliz some inequalities I recently obtained partly together with other authors. The paper is organized in the following way: In section 1 we collect some necessary notations and definitions. We also present some of the classical inequalities we will generalize, unify and complement later on. In section 2 we prove two general “setvalued” inequalities, which in particular generalize some results recently obtained by J. Peetre and the present author [25]. The singlevalued versions of these inequalities are studied in section 3 and several applications are pointed out e.g. a recent result by J. Matkowski [22]. In section 4 we present a generalization of Holder’s inequality to the case with a family of spaces, which recently has been obtained by L. Nikolova and the present author [24]. We also include a generalized completely symmetric form of Holder’s inequality, thereby generalizing some previous results by Aczel-Beckenbach [1] and the present author [29]. In section 5 we prove a sharp generalized form of Minkowski’s inequality. For the proof of this inequality we need to prove a certain generalization of Clarcson’s inequality, which can be of independent interest. Section 6 is used to prove some relations between generalized versions of some classical inequalities. In particular we find that some of these inequalities, in a sense, are equivalent. In section 7 we present a recent result by T. Koski and the present author [15], where in particular the sharpest possible upper bounds for the exponential entropies E(α;f) are obtained for every α > 0, α ≠ 1. In particular by letting α → 1 we obtain a quite new proof of the “differential entropy inequality”, which is one of the corner stones in Information Theory. In section 8 we give some concluding remarks. In particular we present a new results concerning the best constant in a variant of Hardy’s inequality. We also include some remarks concerning recent development of the theory and applications of generalized Gini means. In particular we point out some examples of inequalities which may be seen as limiting cases of classical inequalities.