scholarly journals New Estimations for Shannon and Zipf–Mandelbrot Entropies

Entropy ◽  
2018 ◽  
Vol 20 (8) ◽  
pp. 608 ◽  
Author(s):  
Muhammad Khan ◽  
Zaid Al-sahwi ◽  
Yu-Ming Chu
Keyword(s):  
Convex Function ◽  
Shannon Entropy ◽  
Jensen Inequality

The main purpose of this paper is to find new estimations for the Shannon and Zipf–Mandelbrot entropies. We apply some refinements of the Jensen inequality to obtain different bounds for these entropies. Initially, we use a precise convex function in the refinement of the Jensen inequality and then tamper the weight and domain of the function to obtain general bounds for the Shannon entropy (SE). As particular cases of these general bounds, we derive some bounds for the Shannon entropy (SE) which are, in fact, the applications of some other well-known refinements of the Jensen inequality. Finally, we derive different estimations for the Zipf–Mandelbrot entropy (ZME) by using the new bounds of the Shannon entropy for the Zipf–Mandelbrot law (ZML). We also discuss particular cases and the bounds related to two different parametrics of the Zipf–Mandelbrot entropy. At the end of the paper we give some applications in linguistics.

scholarly journals Estimation of different entropies via Lidstone polynomial using Jensen-type functionals

2020 ◽  
Vol 9 (3) ◽  
pp. 613-631
Author(s):  
Khuram Ali Khan ◽  
Tasadduq Niaz ◽  
Đilda Pečarić ◽  
Josip Pečarić
Keyword(s):  
Convex Function ◽  
Shannon Entropy ◽  
Rényi Divergence ◽  
Lidstone Polynomial

Abstract In this work, some new functional of Jensen-type inequalities are constructed using Shannon entropy, f-divergence, and Rényi divergence, and some estimates are obtained for these new functionals. Also using the Zipf–Mandelbrot law and hybrid Zipf–Mandelbrot law, we investigate some bounds for these new functionals. Furthermore, we generalize these new functionals for m-convex function using Lidstone polynomial.

scholarly journals Strong h-Convexity and Separation Theorems

2016 ◽  
Vol 2016 ◽  
pp. 1-5 ◽  
Author(s):  
Teodoro Lara ◽  
Nelson Merentes ◽  
Kazimierz Nikodem
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Stability Result ◽  
Jensen Inequality ◽  
Separation Theorems

Jensen inequality for strongly h-convex functions and a characterization of pairs of functions that can be separated by a strongly h-convex function are presented. As a consequence, a stability result of the Hyers-Ulam type is obtained.

scholarly journals EXACT UPPER AND LOWER BOUNDS ON THE DIFFERENCE BETWEEN THE ARITHMETIC AND GEOMETRIC MEANS

2015 ◽  
Vol 92 (1) ◽  
pp. 149-158 ◽  
Author(s):  
IOSIF PINELIS
Keyword(s):  
Convex Function ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Jensen Inequality ◽  
Geometric Means ◽  
The Difference

Exact upper and lower bounds on the difference between the arithmetic and geometric means are obtained. The inequalities providing these bounds may be viewed, respectively, as a reverse Jensen inequality and an improvement of the direct Jensen inequality, in the case when the convex function is the exponential.

Estimation of different entropies via Abel–Gontscharoff Green functions and Fink’s identity using Jensen type functionals

2018 ◽  
Vol 26 (1/2) ◽  
pp. 15-39
Author(s):  
Khuram Ali Khan ◽  
Tasadduq Niaz ◽  
Đilda Pečarić ◽  
Josip Pečarić
Keyword(s):  
Convex Function ◽  
Shannon Entropy ◽  
Green Functions ◽  
Fink’S Identity ◽  
New Inequalities

In this work, we estimated the different entropies like Shannon entropy, Rényi divergences, Csiszár divergence by using Jensen’s type functionals. The Zipf’s–Mandelbrot law and hybrid Zipf’s–Mandelbrot law are used to estimate the Shannon entropy. The Abel–Gontscharoff Green functions and Fink’s Identity are used to construct new inequalities and generalized them for m-convex function.

Generalization of cyclic refinements of Jensen inequality by montgomery identity and Green’s function

2018 ◽  
Vol 11 (04) ◽  
pp. 1850060 ◽  
Author(s):  
Nasir Mehmood ◽  
Saad Ihsan Butt ◽  
Josip Pečarić
Keyword(s):  
Convex Function ◽  
Green’S Function ◽  
Green's Function ◽  
Convex Functions ◽  
Mean Value ◽  
Higher Order ◽  
Jensen Inequality ◽  
Linear Functionals ◽  
Montgomery Identity ◽  
Exponential Convexity

We consider discrete and continuous cyclic refinements of Jensen’s inequality and generalize them from convex function to higher order convex function by means of Lagrange Green’s function and Montgomery identity. We give application of our results by formulating the monotonicity of the linear functionals obtained from generalized identities utilizing the theory of inequalities for [Formula: see text]-convex functions at a point. We compute Grüss and Ostrowski type bounds for generalized identities associated with the obtained inequalities. Finally, we investigate the properties of linear functionals regarding exponential convexity log convexity and mean value theorems.

TRACE JENSEN INEQUALITY FOR SELF-ADJOINT OPERATORS IN SEMI-FINITE VON NEUMANN ALGEBRAS

2013 ◽  
Vol 24 (09) ◽  
pp. 1350075
Author(s):  
HIDEKI KOSAKI
Keyword(s):  
Convex Function ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Jensen Inequality ◽  
Measurable Operator ◽  
Von Neumann ◽  
Finite Von Neumann Algebra ◽  
Measurable Operators ◽  
Adjoint Operators ◽  
Finite Von Neumann Algebras

Let [Formula: see text] be a semi-finite von Neumann algebra equipped with a faithful semi-finite normal trace τ, and f(t) be a convex function with f(0) = 0. The trace Jensen inequality in our previous work states τ(f(a*xa)) ≤ τ(a*f(x)a) (as long as the both sides are well-defined) for a contraction [Formula: see text] and a semi-bounded τ-measurable operator x. Validity of this inequality for (not necessarily semi-bounded) self-adjoint τ-measurable operators is investigated.

scholarly journals Integral Inequalities Involving Strongly Convex Functions

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Ying-Qing Song ◽  
Muhammad Adil Khan ◽  
Syed Zaheer Ullah ◽  
Yu-Ming Chu
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Jensen’S Inequality ◽  
Integral Inequalities ◽  
Jensen Inequality ◽  
Jensen's Inequality ◽  
Strongly Convex Functions ◽  
Strongly Convex Function ◽  
Strongly Convex

We study the notions of strongly convex function as well as F-strongly convex function. We present here some new integral inequalities of Jensen’s type for these classes of functions. A refinement of companion inequality to Jensen’s inequality established by Matić and Pečarić is shown to be recaptured as a particular instance. Counterpart of the integral Jensen inequality for strongly convex functions is also presented. Furthermore, we present integral Jensen-Steffensen and Slater’s inequality for strongly convex functions.

scholarly journals Refinements of Jensen’s Inequality via Majorization Results with Applications in the Information Theory

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Yongping Deng ◽  
Hidayat Ullah ◽  
Muhammad Adil Khan ◽  
Sajid Iqbal ◽  
Shanhe Wu
Keyword(s):  
Information Theory ◽  
Shannon Entropy ◽  
Probability Distributions ◽  
Jensen’S Inequality ◽  
Jensen Inequality ◽  
Jensen's Inequality ◽  
Power Means ◽  
Quasiarithmetic Means

In this study, we present some new refinements of the Jensen inequality with the help of majorization results. We use the concept of convexity along with the theory of majorization and obtain refinements of the Jensen inequality. Moreover, as consequences of the refined Jensen inequality, we derive some bounds for power means and quasiarithmetic means. Furthermore, as applications of the refined Jensen inequality, we give some bounds for divergences, Shannon entropy, and various distances associated with probability distributions.

Extreme-point solutions in Markov decision processes

1983 ◽  
Vol 20 (04) ◽  
pp. 835-842
Author(s):  
David Assaf
Keyword(s):  
Convex Function ◽  
Extreme Point ◽  
Markov Decision Processes ◽  
Convex Functions ◽  
Sufficient Conditions ◽  
Decision Processes ◽  
Markov Decision ◽  
Full Solution

The paper presents sufficient conditions for certain functions to be convex. Functions of this type often appear in Markov decision processes, where their maximum is the solution of the problem. Since a convex function takes its maximum at an extreme point, the conditions may greatly simplify a problem. In some cases a full solution may be obtained after the reduction is made. Some illustrative examples are discussed.

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