Refinement of Discrete Lah–Ribarič Inequality and Applications on Csiszár Divergence

Đilda Pečarić,Josip Pečarić,Jurica Perić

doi:10.3390/math10050755

Đilda Pečarić, Josip Pečarić + Show 1 more

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https://doi.org/10.3390/math10050755

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Abstract

In this paper we give a new refinement of the Lah–Ribarič inequality and, using the same technique, we give a refinement of the Jensen inequality. Using these results, a refinement of the discrete Hölder inequality and a refinement of some inequalities for discrete weighted power means and discrete weighted quasi-arithmetic means are obtained. We also give applications in the information theory; namely, we give some interesting estimations for the discrete Csiszár divergence and for its important special cases.

Highlights

Research of the classical inequalities, such as the Jensen, the Hölder and similar, has experienced great expansion
Let I be an interval in R and f : I → R a convex function
If we look at the previous proof, we see that the technique is the same as for our main result and the refinement of the Jensen inequality

Summary

Introduction

Research of the classical inequalities, such as the Jensen, the Hölder and similar, has experienced great expansion These inequalities first appeared in discrete and integral forms, and many generalizations and improvements have been proved (for instance, see [1,2]). Pn ) a nonnegative n-tuple such that Pn = ∑in=1 pi > 0, the well known Jensen’s inequality. Jensen’s inequality is one of the most famous inequalities in convex analysis, for which special cases are other well-known inequalities (such as Hölder’s inequality, A-G-H inequality, etc.). Beside mathematics, it has many applications in statistics, information theory, and engineering.

New Refinements

Applications in Information Theory

Conclusions