Generalization of the Levinson inequality with applications to information theory

Abstract

In the presented paper, Levinson's inequality for the 3-convex function is generalized by using two Green functions.Čebyšev-, Grüss-and Ostrowski-type new bounds are found for the functionals involving data points of two types. Moreover, the main results are applied to information theory via the f -divergence, the Rényi divergence, the Rényi entropy, the Shannon entropy and the Zipf-Mandelbrot law.

Levinson type inequalities for higher order convex functions via Abel–Gontscharoff interpolation

“…In [8], Adeel et al generalized Levinson's inequality for 3-convex function by using two Green functions. In [9], Pečarić et al gave a probabilistic version of Levinson's inequality (2) under Mercer's assumption of equal variances (but for a different number of data points) for the family of 3-convex functions at a point.…”

“…Theorem 14 Assume H. Let f ∈ C m [0, 2a] (m ≥ 3) with f (m-1) be absolutely continuous. Also let G F,m andJ (f (•)) be defined in (8) and (17) respectively. If…”

“…Let f : I 1 = [ζ 1 , ζ 2 ] → R be such that f ∈ C m [ζ 1 , ζ 2 ] (m ≥ 3) and f (m-1) is absolutely continuous. Also let G F,m andJ (f (•)) be defined in(8) and(12) respectively. If(14) is valid, then (15) is also valid.Proof Proof is similar to Theorem 8.Let f ∈ C m [ζ 1 , ζ 2 ] (m ≥ 3), p = (p 1 , .…”

“…Let f ∈ C m [ζ 1 , ζ 2 ] (m ≥ 3) and f (m) be absolutely continuous with (.ζ 1 )(ζ 2 -•)[f (m+1) ] 2 ∈ L[ζ 1 , ζ 2 ]. Also let G F,m andJ (f (•)) as defined in(8) and (12) respectively. Then we havȇ…”

Shannon type inequalities via time scales theory

“…In recent times, Shannon entropy and Zipf-Mandelbrot law have been the topics of great interest, see for example [1,9,11,12]. The concept of Shannon entropy, the central source of information theory, is sometimes referred to as measure of uncertainty.…”