Abstract
In this paper we give extensions of Sherman’s inequality considering the class of convex functions of higher order. As particular cases, we get an extended weighted majorization inequality as well as Jensen’s inequality which have direct connection to information theory. We use the obtained results to derive new estimates for Shannon’s and Rényi’s entropy, information energy, and some well-known measures between probability distributions. Using the Zipf–Mandelbrot law, we introduce new functionals to derive some related results.
Highlights
If we substitute1 pi in place of ξi in (4.1), we get the required result. Applying discrete Jensen’s inequality to the convex function φ(x) = – ln x, we have m m ln pixi ≥ pi ln xi
1 Introduction and preliminaries We start with a brief overview of divided differences and n-convex functions and give some basic results from the majorization theory
[x0, x1, . . . , xn; φ] ≥ 0 holds for all choices of (n + 1) distinct points xi ∈ [α, β], i = 0, . . . , n
Summary
1 pi in place of ξi in (4.1), we get the required result. Applying discrete Jensen’s inequality to the convex function φ(x) = – ln x, we have m m ln pixi ≥ pi ln xi. Proof Substituting pλi –1 in place of ξi in (4.1), we obtain the required result. (ii) Substituting ξi in place of xi, pi in place of ai in (3.10), and choosing φ(x) = xλ, λ ∈ (1, ∞), we obtain the required result. Proof (i) Substituting pi in place of ξi in (4.2) and taking into account that m pλi = exp (1 – λ)Hλ(X) , i=1 i.e., m pλi +1 = exp –λHλ+1(X) , i=1 we get (4.4). We consider positive probability distributions u, v, w with the assumption of existence of a row stochastic matrix A ∈ Mm(R) such that w = wA and v = u AT ,. Definition 1 (Csiszár divergence for Z–M law) Let m ∈ N and φ : [α, β] → R be a function.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
