Majorization, “useful” Csiszár divergence and “useful” Zipf-Mandelbrot law

Abstract

In this paper, we consider the definition of “useful” Csiszár divergence and “useful” Zipf-Mandelbrot law associated with the real utility distribution to give the results for majorizatioQn inequalities by using monotonic sequences. We obtain the equivalent statements between continuous convex functions and Green functions via majorization inequalities, “useful” Csiszár functional and “useful” Zipf-Mandelbrot law. By considering “useful” Csiszár divergence in the integral case, we give the results for integral majorization inequality. Towards the end, some applications are given.

A New Bound for the Jensen Gap With Applications in Information Theory

“…For deriving the main result, we need the following Green function defined on [ω 1 , ω 2 ] × [ω 1 , ω 2 ] [22]:…”

New refinement of the Jensen inequality associated to certain functions with applications

“…In this section, we present some important applications for different divergences and distances in information theory [22] of our main result.…”

Converses of the Jensen inequality derived from the Green functions with applications in information theory

“…Definition (Csiszár divergence): Let be a subset of the set of nonnegative real numbers and be a function. Also, let be two probability density functions such that for all then the Csiszár divergence is defined as …”

“…(Kullback-Leibler divergence): For two positive probability densities U(z) and V(z) defined on [b 1 , b 2 ], the Kullback-Leibler divergence is defined as33 …”

“…(Csiszár divergence): Let [ 1 , 2 ] be a subset of the set of nonnegative real numbers and T ∶ [1 , 2 ] → R be a function. Also, let U ∶ [b 1 , b 2 ] → [ 1 , 2 ], V ∶ [b 1 , b 2 ] → (0, ∞) be two probability density functions such that ∈ [ 1 , 2 ] for all z ∈ [b 1 , b 2 ], then the Csiszár divergence is defined as33 …”