Abstract
Jensen’s inequality is one of the fundamental inequalities which has several applications in almost every field of science. In 2003, Mercer gave a variant of Jensen’s inequality which is known as Jensen–Mercer’s inequality. The purpose of this article is to propose new bounds for Csiszár and related divergences by means of Jensen–Mercer’s inequality. Also, we investigate several new bounds for Zipf–Mandelbrot entropy. The idea of this article may further stimulate research on information theory with the help of Jensen–Mercer’s inequality.
Highlights
In theory of inequalities, convex functions play an important role. e definition of convex function [1] is as follows.Let φ: [δ, ε] ⊂ R ⟶ R be a function, φ is said to be convex if ∀ w, z ∈ [δ, ε] and 0 ≤ t ≤ 1, the inequality φ(tw +(1 − t)z) ≤ tφ(w) +(1 − t)φ(z), (1)holds, and φ is said to be strictly convex if ∀ w ≠ z and t ∈ (0, 1), (1) holds strictly
Holds, and φ is said to be strictly convex if ∀ w ≠ z and t ∈ (0, 1), (1) holds strictly
If inequality (1) holds in reversed directions, φ is said to be concave, and φ is said to be strictly concave if the inequality (1) holds strictly in reversed direction ∀ w ≠ z and t ∈ (0, 1)
Summary
Convex functions play an important role. e definition of convex function [1] is as follows. Kullback–Leibler divergence is nonnegative and is zero if and only if θλ cλ It satisfies the two properties of metric, but K(θ, c) ≠ K(c, θ), and does not obey the triangle inequality. We can construct Shannon entropy [37] from Kullback–Leibler divergence, which is given as follows: H(θ). Variational distance is utilized to characterize solid typicality and asymptotic equipartition of sequences generated by sampling from a given distribution [48]. Zipf’s law, known as Zipf–Mandelbrot law, which gave improvement on account of the low-rank words in corpus [58] and is given as follows:. Ey have used two parametric Zipf–Mandelbrot laws instead of different weights in the inequalities for Shannon entropy. We establish bounds for Zipf–Mandelbrot entropy by applying Zipf–Mandelbrot laws instead of probability distributions in Kullback–Leibler and Jefferys divergences. We deduce new estimates for Zipf–Mandelbrot entropy associated to different parametric Zipf–Mandelbrot laws
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