scholarly journals GENERALIZATION AND REFINEMENTS OF JENSEN INEQUALITY

2021 ◽  
Vol 12 (5) ◽  
pp. 1-27
Author(s):  
Faiza Rubab ◽  
Hira Nabi ◽  
Asif R. Khan
Keyword(s):  
Information Theory ◽  
Upper Bound ◽  
Jensen Inequality ◽  
Integral Version

We give generalizations and refinements of Jensen and Jensen− Mercer inequalities by using weights which satisfy the conditions of Jensen and Jensen− Steffensen inequalities. We also give some refinements for discrete and integral version of generalized Jensen−Mercer inequality and shown to be an improvement of the upper bound for the Jensen’s difference given in [32]. Applications of our work include new bounds for some important inequalities used in information theory, and generalizing the relations among means.

scholarly journals A New Information Inequality and Its Application in Establishing Relation Among Various f-Divergence Measures

2012 ◽  
Vol 8 (1) ◽  
pp. 17-32 ◽  
Author(s):  
K. Jain ◽  
Ram Saraswat
Keyword(s):  
Information Theory ◽  
Jensen Inequality ◽  
Chi Square ◽  
Information Inequality ◽  
Divergence Measures ◽  
New Information

A New Information Inequality and Its Application in Establishing Relation Among Various f-Divergence MeasuresAn Information inequality by using convexity arguments and Jensen inequality is established in terms of Csiszar f-divergence measures. This inequality is applied in comparing particular divergences which play a fundamental role in Information theory, such as Kullback-Leibler distance, Hellinger discrimination, Chi-square distance, J-divergences and others.

A New Refinement of the Jensen Inequality with Applications in Information Theory

2020 ◽  
Vol 44 (1) ◽  
pp. 267-278 ◽  
Author(s):  
Muhammad Adil Khan ◽  
Ɖilda Pečarić ◽  
Josip Pečarić
Keyword(s):  
Information Theory ◽  
Jensen Inequality

Laser-induced addition of hydrogen fluoride to unsaturated molecules: the HF + CH2CF2 system

1984 ◽  
Vol 62 (11) ◽  
pp. 2302-2309 ◽  
Author(s):  
Walter H. Beck ◽  
George Burns
Keyword(s):  
Mass Spectrometry ◽  
Information Theory ◽  
Gas Chromatography ◽  
Gas Phase ◽  
Hydrogen Fluoride ◽  
Vibrational Level ◽  
Upper Bound ◽  
Addition Reactions ◽  
Reaction Products ◽  
Addition Rate

Although certain classes of reactions are known to occur via 4-centre transition states, there has been no irrefutable proof that in the gas phase addition reactions also proceed via a 4-centre concerted process. Results of recent information theory calculations on the HF(υ) + CH2CF2 system indicate that it should be possible to observe reaction products for HF(υ = 5). Results are presented here on experiments attempting to achieve laser-induced 4-centre addition of HF to several unsaturated molecules (C2H2, C2H4, C4H6, CH2CF2, and CF2CClF). A mixture of HF and co-reactant was irradiated by a dye-laser tuned photo-acoustically to the fifth vibrational level of HF, and the resulting mixture was analysed by gas chromatography and mass spectrometry. There was no evidence of any laser-induced products. An upper bound was deduced for the HF + CH2CF2 addition rate constant at 610 K: [Formula: see text].

scholarly journals Reversing Jensen’s Inequality for Information-Theoretic Analyses

Information ◽  
2022 ◽  
Vol 13 (1) ◽  
pp. 39
Author(s):  
Neri Merhav
Keyword(s):  
Information Theory ◽  
Jensen’S Inequality ◽  
Jensen Inequality ◽  
Concave Functions ◽  
Jensen's Inequality ◽  
Information Theoretic ◽  
Main Ideas

In this work, we propose both an improvement and extensions of a reverse Jensen inequality due to Wunder et al. (2021). The new proposed inequalities are fairly tight and reasonably easy to use in a wide variety of situations, as demonstrated in several application examples that are relevant to information theory. Moreover, the main ideas behind the derivations turn out to be applicable to generate bounds to expectations of multivariate convex/concave functions, as well as functions that are not necessarily convex or concave.

scholarly journals Extrapolation of convex functions

Filomat ◽  
2018 ◽  
Vol 32 (1) ◽  
pp. 127-139
Author(s):  
M. Sababheh
Keyword(s):  
Lower Bound ◽  
Convex Hull ◽  
Upper Bound ◽  
Convex Functions ◽  
Jensen Inequality ◽  
Matlab Code

The idea of the well known Jensen inequality is to interpolate convex functions, by finding an upper bound of the function at a point in the convex hull of predefined points. In this article, we present a counterpart of this inequality by giving a lower bound of the function outside this convex hull. This inequality is then refined by finding as many refining positive terms as we wish. Some applications treating means, integrals and eigenvalues are given in the end. Moreover, we present a MATLAB code that helps generate the parameters appearing in our results.

scholarly journals Universal constraints on selection strength in lineage trees

2020 ◽  
Author(s):  
Arthur Genthon ◽  
David Lacoste
Keyword(s):  
Natural Selection ◽  
Upper Bound ◽  
Fitness Landscape ◽  
Time Lapse ◽  
Jensen Inequality ◽  
Phenotypic Trait ◽  
Stochastic Thermodynamics ◽  
Growing Cell ◽  
The Difference ◽  
Lineage Trees

We obtain general inequalities constraining the difference between the average of an arbitrary function of a phenotypic trait, which includes the fitness landscape of the trait itself, in the presence or in the absence of natural selection. These inequalities imply bounds on the strength of selection, which can be measured from the statistics of traits or divisions along lineages. The upper bound is related to recent generalizations of linear response relations in Stochastic Thermodynamics, and is reminiscent of the fundamental theorem of Natural selection of R. Fisher and of its generalization by Price. The lower bound follows from recent improvements on Jensen inequality and is typically less tight than the upper bound. We illustrate our results using numerical simulations of growing cell colonies and with experimental data of time-lapse microscopy experiments of bacteria cell colonies.

scholarly journals Extensions of recent combinatorial refinements of discrete and integral Jensen inequalities

2021 ◽  
Author(s):  
László Horváth
Keyword(s):  
Information Theory ◽  
Geometric Mean ◽  
Arithmetic Mean ◽  
Jensen Inequality ◽  
Arithmetic Means ◽  
Norm Inequalities ◽  
Integrable Functions ◽  
Essential Extensions ◽  
Vector Valued

AbstractThe main purpose of this work is to present essential extensions of results in [7] and [8], and apply them to some special situations. Of particular interest is the refinement of the integral Jensen inequality for vector valued integrable functions. The applications related to four topics, namely f-divergences in information theory (an interesting refinement of the weighted geometric mean–arithmetic mean inequality is obtained as a consequence), norm inequalities, quasi-arithmetic means, Hölder’s and Minkowski’s inequalities.

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.

scholarly journals Determination of Bounds for the Jensen Gap and Its Applications

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 3132
Author(s):  
Hidayat Ullah ◽  
Muhammad Adil Khan ◽  
Tareq Saeed
Keyword(s):  
Information Theory ◽  
Current Literature ◽  
Convex Functions ◽  
Concave Function ◽  
Hölder Inequality ◽  
Jensen Inequality ◽  
Power Mean ◽  
Arithmetic Means ◽  
Jensen Difference

The Jensen inequality has been reported as one of the most consequential inequalities that has a lot of applications in diverse fields of science. For this reason, the Jensen inequality has become one of the most discussed developmental inequalities in the current literature on mathematical inequalities. The main intention of this article is to find some novel bounds for the Jensen difference while using some classes of twice differentiable convex functions. We obtain the proposed bounds by utilizing the power mean and Höilder inequalities, the notion of convexity and the prominent Jensen inequality for concave function. We deduce several inequalities for power and quasi-arithmetic means as a consequence of main results. Furthermore, we also establish different improvements for Hölder inequality with the help of obtained results. Moreover, we present some applications of the main results in information theory.

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