scholarly journals Bounds for the Jensen Gap in terms of Power Means with Applications

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Xuexiao You ◽  
Muhammad Adil Khan ◽  
Hamid Reza Moradi
Keyword(s):  
Jensen’S Inequality ◽  
Jensen's Inequality ◽  
Hadamard Inequality ◽  
Quantum Calculus ◽  
Power Means ◽  
Quantum Integral ◽  
The Difference ◽  
Almost All ◽  
Two Sides ◽  
Integral Version

Jensen’s and its related inequalities have attracted the attention of several mathematicians due to the fact that Jensen’s inequality has numerous applications in almost all disciplines of mathematics and in other fields of science. In this article, we propose new bounds for the difference of two sides of Jensen’s inequality in terms of power means. An example has been presented for the importance and support of the main results. Related results have been given in quantum calculus. As consequences, improvements of quantum integral version of Hermite-Hadamard inequality have been derived. The obtained inequalities have been applied for some well-known inequalities such as Hermite-Hadamrd, Hölder, and power mean inequalities. Finally, some applications are given in information theory. The tools performed for obtaining the main results may be applied to obtain more results for other inequalities.

scholarly journals Generalized Jensen-Mercer Inequality for Functions with Nondecreasing Increments

2016 ◽  
Vol 2016 ◽  
pp. 1-12
Author(s):  
Asif R. Khan ◽  
Sumayyah Saadi
Keyword(s):  
Convex Functions ◽  
Jensen’S Inequality ◽  
Higher Dimensions ◽  
Jensen's Inequality ◽  
Valuable Addition ◽  
Interesting Variation ◽  
Integral Version ◽  
The Way

In the year 2003, McD Mercer established an interesting variation of Jensen’s inequality and later in 2009 Mercer’s result was generalized to higher dimensions by M. Niezgoda. Recently, Asif et al. has stated an integral version of Niezgoda’s result for convex functions. We further generalize Niezgoda’s integral result for functions with nondecreasing increments and give some refinements with applications. In the way, we generalize an important result, Jensen-Boas inequality, using functions with nondecreasing increments. These results would constitute a valuable addition to Jensen-type inequalities in the literature.

Cyclic Refinements of the Different Versions of Operator Jensen's Inequality

2016 ◽  
Vol 31 ◽  
pp. 125-133 ◽  
Author(s):  
Laszlo Horvath ◽  
Khuram Khan ◽  
Josip Pecaric
Keyword(s):  
Convex Functions ◽  
Jensen’S Inequality ◽  
Jensen's Inequality ◽  
Norm Inequalities ◽  
Power Means ◽  
Operator Convex ◽  
Operator Convex Functions

Refinements of the operator Jensen's inequality for convex and operator convex functions are given by using cyclic refinements of the discrete Jensen's inequality. Similar refinements are fairly rare in the literature. Some applications of the results to norm inequalities, the Holder McCarthy inequality and generalized weighted power means for operators are presented.

scholarly journals Estimation of divergence measures via weighted Jensen inequality on time scales

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Iqrar Ansari ◽  
Khuram Ali Khan ◽  
Ammara Nosheen ◽  
Ðilda Pečarić ◽  
Josip Pečarić
Keyword(s):  
Time Scales ◽  
Lower Bounds ◽  
Time Scale ◽  
Jensen’S Inequality ◽  
Jensen Inequality ◽  
Jensen's Inequality ◽  
Quantum Calculus ◽  
Divergence Measures ◽  
Zipf Law ◽  
New Inequalities

AbstractThe main purpose of the presented paper is to obtain some time scale inequalities for different divergences and distances by using weighted time scales Jensen’s inequality. These results offer new inequalities in h-discrete calculus and quantum calculus and extend some known results in the literature. The lower bounds of some divergence measures are also presented. Moreover, the obtained discrete results are given in the light of the Zipf–Mandelbrot law and the Zipf law.

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.

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 Refinements of the integral Jensen’s inequality generated by finite or infinite permutations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
László Horváth
Keyword(s):  
Jensen’S Inequality ◽  
The Other ◽  
New Method ◽  
Jensen's Inequality ◽  
Arithmetic Means ◽  
Hadamard Inequality ◽  
Other Hand ◽  
Significant Generalization ◽  
Fejér Inequality ◽  
The One

AbstractThere are a lot of papers dealing with applications of the so-called cyclic refinement of the discrete Jensen’s inequality. A significant generalization of the cyclic refinement, based on combinatorial considerations, has recently been discovered by the author. In the present paper we give the integral versions of these results. On the one hand, a new method to refine the integral Jensen’s inequality is developed. On the other hand, the result contains some recent refinements of the integral Jensen’s inequality as elementary cases. Finally some applications to the Fejér inequality (especially the Hermite–Hadamard inequality), quasi-arithmetic means, and f-divergences are presented.

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 Reverses of the Jensen-Type Inequalities for Signed Measures

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Rozarija Jakšić ◽  
Josip Pečarić ◽  
Mirna Rodić Lipanović
Keyword(s):  
Jensen’S Inequality ◽  
Cauchy Type ◽  
Jensen's Inequality ◽  
Logarithmic Convexity ◽  
Left Hand ◽  
New Class ◽  
Right Hand ◽  
The Difference ◽  
The Right

In this paper we derive refinements of the Jensen type inequalities in the case of real Stieltjes measuredλ, not necessarily positive, which are generalizations of Jensen's inequality and its reverses for positive measures. Furthermore, we investigate the exponential and logarithmic convexity of the difference between the left-hand and the right-hand side of these inequalities and give several examples of the families of functions for which the obtained results can be applied. The outcome is a new class of Cauchy-type means.

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