An Extension of Young's Inequality

Young's inequality is extended to the context of absolutely continuous measures. Several applications are included.Comment: 15 pages, 6 figure

On some properties of Tsallis hypoentropies and hypodivergences

Both the Kullback-Leibler and the Tsallis divergence have a strong limitation: if the value $0$ appears in probability distributions $\left( p_{1},\cdots ,p_{n}\right)$ and $\left( q_{1},\cdots ,q_{n}\right)$, it must appear in the same positions for the sake of significance. In order to avoid that limitation in the framework of Shannon statistics, Ferreri introduced in 1980 the hypoentropy: "such conditions rarely occur in practice". The aim of the present paper is to extend Ferreri's hypoentropy to the Tsallis statistics. We introduce the Tsallis hypoentropy and the Tsallis hypodivergence and describe their mathematical behavior. Fundamental properties like nonnegativity, monotonicity, the chain rule and subadditivity are established.Comment: 23 page

Structural results on convexity relative to cost functions

Mass transportation problems appear in various areas of mathematics, their solutions involving cost convex potentials. Fenchel duality also represents an important concept for a wide variety of optimization problems, both from the theoretical and the computational viewpoints. We drew a parallel to the classical theory of convex functions by investigating the cost convexity and its connections with the usual convexity. We give a generalization of Jensen's inequality for cost convex functions.Comment: 10 page

Some inequalities on generalized entropies

We give several inequalities on generalized entropies involving Tsallis entropies, using some inequalities obtained by improvements of Young's inequality. We also give a generalized Han's inequality.Comment: 15 page

On the Jensen functional and superterzaticity

In this note we describe some results concerning upper and lower bounds for the Jensen functional. We use several known and new results to shed light on the concept of superterzatic function. Particular cases of interest are also considered. Keywords: Jensen functional, Superterzatic function