scholarly journals Uniform Treatment of Jensen’s Inequality by Montgomery Identity

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
Tahir Rasheed ◽  
Saad Ihsan Butt ◽  
Đilda Pečarić ◽  
Josip Pečarić ◽  
Ahmet Ocak Akdemir
Keyword(s):  
Information Theory ◽  
Integral Inequality ◽  
Convex Functions ◽  
Jensen’S Inequality ◽  
Fractional Integrals ◽  
Jensen's Inequality ◽  
Hadamard Inequality ◽  
Montgomery Identity ◽  
Discrete Inequality ◽  
Uniform Treatment

We generalize Jensen’s integral inequality for real Stieltjes measure by using Montgomery identity under the effect of n − convex functions; also, we give different versions of Jensen’s discrete inequality along with its converses for real weights. As an application, we give generalized variants of Hermite–Hadamard inequality. Montgomery identity has a great importance as many inequalities can be obtained from Montgomery identity in q − calculus and fractional integrals. Also, we give applications in information theory for our obtained results, especially for Zipf and Hybrid Zipf–Mandelbrot entropies.

On Some Integral Inequalities Related to Hardy's

1977 ◽  
Vol 20 (3) ◽  
pp. 307-312 ◽  
Author(s):  
Christopher Olutunde Imoru
Keyword(s):  
Integral Inequality ◽  
Convex Functions ◽  
Jensen’S Inequality ◽  
Integral Inequalities ◽  
Hardy’S Inequality ◽  
Jensen's Inequality ◽  
Hardy's Inequality ◽  
Special Case

AbstractWe obtain mainly by using Jensen's inequality for convex functions an integral inequality, which contains as a special case Shun's generalization of Hardy's inequality.

scholarly journals Refinements of Jensen's inequality and applications

2022 ◽  
Vol 7 (4) ◽  
pp. 5328-5346
Author(s):  
Tareq Saeed ◽  
◽  
Muhammad Adil Khan ◽  
Hidayat Ullah ◽  
Keyword(s):  
Information Theory ◽  
Shannon Entropy ◽  
Research Work ◽  
Jensen’S Inequality ◽  
Hölder Inequality ◽  
Jensen's Inequality ◽  
Arithmetic Means ◽  
Hadamard Inequality ◽  
Bhattacharyya Coefficient

<abstract><p>The principal aim of this research work is to establish refinements of the integral Jensen's inequality. For the intended refinements, we mainly use the notion of convexity and the concept of majorization. We derive some inequalities for power and quasi–arithmetic means while utilizing the main results. Moreover, we acquire several refinements of Hölder inequality and also an improvement of Hermite–Hadamard inequality as consequences of obtained results. Furthermore, we secure several applications of the acquired results in information theory, which consist bounds for Shannon entropy, different divergences, Bhattacharyya coefficient, triangular discrimination and various distances.</p></abstract>

scholarly journals Application of Functionals in Creating Inequalities

2016 ◽  
Vol 2016 ◽  
pp. 1-6 ◽  
Author(s):  
Zlatko Pavić ◽  
Shanhe Wu ◽  
Vedran Novoselac
Keyword(s):  
Convex Functions ◽  
Functional Form ◽  
Closed Interval ◽  
Jensen’S Inequality ◽  
Linear Functionals ◽  
Jensen's Inequality ◽  
Hadamard Inequality ◽  
Positive Linear Functionals ◽  
Main Inequality

The paper deals with the fundamental inequalities for convex functions in the bounded closed interval. The main inequality includes convex functions and positive linear functionals extending and refining the functional form of Jensen’s inequality. This inequality implies the Jensen, Fejér, and, thus, Hermite-Hadamard inequality, as well as their refinements.

scholarly journals SUPERADDITIVITY OF SOME FUNCTIONALS ASSOCIATED WITH JENSEN’S INEQUALITY FOR CONVEX FUNCTIONS ON LINEAR SPACES WITH APPLICATIONS

2010 ◽  
Vol 82 (1) ◽  
pp. 44-61 ◽  
Author(s):  
S. S. DRAGOMIR
Keyword(s):  
Information Theory ◽  
Convex Functions ◽  
Convex Sets ◽  
Jensen’S Inequality ◽  
Jensen's Inequality ◽  
Norm Inequalities ◽  
Linear Spaces ◽  
Normed Linear Spaces

AbstractSome new results related to Jensen’s celebrated inequality for convex functions defined on convex sets in linear spaces are given. Applications for norm inequalities in normed linear spaces and f-divergences in information theory are provided as well.

scholarly journals SOME REVERSES OF THE JENSEN INEQUALITY WITH APPLICATIONS

2013 ◽  
Vol 87 (2) ◽  
pp. 177-194 ◽  
Author(s):  
S. S. DRAGOMIR
Keyword(s):  
Information Theory ◽  
Convex Functions ◽  
Hölder’S Inequality ◽  
General Setting ◽  
Jensen’S Inequality ◽  
Jensen Inequality ◽  
Lebesgue Integral ◽  
Jensen's Inequality ◽  
Hölder's Inequality ◽  
Divergence Measures

AbstractTwo new reverses of the celebrated Jensen’s inequality for convex functions in the general setting of the Lebesgue integral, with applications to means, Hölder’s inequality and$f$-divergence measures in information theory, are given.

scholarly journals Generalized Jensen’s functional on time scales via extended Montgomery identity

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Sofia Ramzan ◽  
Ammara Nosheen ◽  
Rabia Bibi ◽  
Josip Pečarić
Keyword(s):  
Time Scales ◽  
Convex Functions ◽  
Jensen’S Inequality ◽  
Jensen's Inequality ◽  
Montgomery Identity

AbstractIn the paper, we use Jensen’s inequality for diamond integrals and generalize it for n-convex functions with the help of an extended Montgomery identity. Moreover, the bounds have been suggested for identities associated with the generalized Jensen-type functional.

scholarly journals Conformable Fractional Integrals Versions of Hermite-Hadamard Inequalities and Their Generalizations

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Muhammad Adil Khan ◽  
Yu-Ming Chu ◽  
Artion Kashuri ◽  
Rozana Liko ◽  
Gohar Ali
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Fractional Integrals ◽  
Montgomery Identity

We prove new Hermite-Hadamard inequalities for conformable fractional integrals by using convex function, s-convex, and coordinate convex functions. We prove new Montgomery identity and by using this identity we obtain generalized Hermite-Hadamard type inequalities.

scholarly journals On a new converse of Jensen's inequality

2009 ◽  
Vol 85 (99) ◽  
pp. 107-110 ◽  
Author(s):  
Slavko Simic
Keyword(s):  
Upper Bound ◽  
Jensen’S Inequality ◽  
Jensen's Inequality ◽  
Discrete Inequality ◽  
Better Than

We give another global upper bound for Jensen's discrete inequality which is better than already existing ones. For instance, we determine a new converses for generalized A-G and G-H inequalities.

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