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 Some Dynamic Inequalities via Diamond Integrals for Function of Several Variables

2021 ◽  
Vol 5 (4) ◽  
pp. 207
Author(s):  
Muhammad Bilal ◽  
Khuram Ali Khan ◽  
Hijaz Ahmad ◽  
Ammara Nosheen ◽  
Khalid Mahmood Awan ◽  
...  
Keyword(s):  
Time Scales ◽  
Time Scale ◽  
Jensen’S Inequality ◽  
Hardy Type Inequalities ◽  
Hardy Type ◽  
Several Variables ◽  
Time Scale Calculus ◽  
Special Cases ◽  
Function Of Several Variables ◽  
New Inequalities

In this paper, Jensen’s inequality and Fubini’s Theorem are extended for the function of several variables via diamond integrals of time scale calculus. These extensions are used to generalize Hardy-type inequalities with general kernels via diamond integrals for the function of several variables. Some Hardy Hilbert and Polya Knop type inequalities are also discussed as special cases. Classical and new inequalities are deduced from the main results using special kernels and particular time scales.

scholarly journals Some inequalities for Csiszár divergence via theory of time scales

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Iqrar Ansari ◽  
Khuram Ali Khan ◽  
Ammara Nosheen ◽  
Ðilda Pečarić ◽  
Josip Pečarić
Keyword(s):  
Time Scales ◽  
Time Scale ◽  
Probability Measures ◽  
Quantum Calculus ◽  
Divergence Measures

AbstractIn this paper, we present some inequalities for Csiszár f-divergence between two probability measures on time scale. These results extend some known results in the literature and offer new results in h-discrete calculus and quantum calculus. We also present several inequalities for divergence measures.

scholarly journals Estimation of divergence measures on time scales via Taylor’s polynomial and Green’s function with applications in q-calculus

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Iqrar Ansari ◽  
Khuram Ali Khan ◽  
Ammara Nosheen ◽  
Ðilda Pečarić ◽  
Josip Pečarić
Keyword(s):  
Green’S Function ◽  
Time Scales ◽  
Green's Function ◽  
Convex Functions ◽  
Higher Order ◽  
Quantum Calculus ◽  
Divergence Measures ◽  
New Inequalities

AbstractTaylor’s polynomial and Green’s function are used to obtain new generalizations of an inequality for higher order convex functions containing Csiszár divergence on time scales. Various new inequalities for some divergence measures in quantum calculus and h-discrete calculus are also established.

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 INEQUALITIES IN TERMS OF THE GÂTEAUX DERIVATIVES FOR CONVEX FUNCTIONS ON LINEAR SPACES WITH APPLICATIONS

2011 ◽  
Vol 83 (3) ◽  
pp. 500-517 ◽  
Author(s):  
S. S. DRAGOMIR
Keyword(s):  
Convex Functions ◽  
Jensen’S Inequality ◽  
Jensen's Inequality ◽  
Linear Spaces ◽  
Divergence Measures ◽  
Gateaux Derivatives ◽  
Gâteaux Derivatives

AbstractSome inequalities in terms of the Gâteaux derivatives related to Jensen’s inequality for convex functions defined on linear spaces are given. Applications for norms, mean f-deviations and f-divergence measures are provided as well.

scholarly journals Inequalities of Hardy Type via Superquadratic Functions with General Kernels and Measures for Several Variables on Time Scales

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
H. M. Rezk ◽  
H. A. Abd El-Hamid ◽  
A. M. Ahmed ◽  
Ghada AlNemer ◽  
M. Zakarya
Keyword(s):  
Time Scales ◽  
Time Scale ◽  
Minkowski Inequality ◽  
Jensen Inequality ◽  
Hardy Type ◽  
Several Variables ◽  
Special Cases ◽  
Superquadratic Functions

We use the properties of superquadratic functions to produce various improvements and popularizations on time scales of the Hardy form inequalities and their converses. Also, we include various examples and interpretations of the disparities in the literature that exist. In particular, our findings can be seen as refinements of some recent results closely linked to the time-scale inequalities of the classical Hardy, Pólya-Knopp, and Hardy-Hilbert. Some continuous inequalities are derived from the main results as special cases. The essential results will be proved by making use of some algebraic inequalities such as the Minkowski inequality, the refined Jensen inequality, and the Bernoulli inequality on time scales.

scholarly journals Novel Fractional Dynamic Hardy–Hilbert-Type Inequalities on Time Scales with Applications

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2964
Author(s):  
Ahmed A. El-Deeb ◽  
Jan Awrejcewicz
Keyword(s):  
Present Article ◽  
Time Scales ◽  
Jensen’S Inequality ◽  
Legendre Transform ◽  
Jensen's Inequality ◽  
Discrete Inequalities ◽  
Special Case ◽  
Hilbert Type

The main objective of the present article is to prove some new ∇ dynamic inequalities of Hardy–Hilbert type on time scales. We present and prove very important generalized results with the help of Fenchel–Legendre transform, submultiplicative functions. We prove the (γ,a)-nabla conformable Hölder’s and Jensen’s inequality on time scales. We prove several inequalities due to Hardy–Hilbert inequalities on time scales. Furthermore, we introduce the continuous inequalities and discrete inequalities as special case.

scholarly journals Some new inequalities via s-convex functions on time scales

2021 ◽  
Vol 13 (1) ◽  
pp. 239-257
Author(s):  
Naila Mehreen ◽  
Matloob Anwar
Keyword(s):  
Convex Function ◽  
Time Scales ◽  
Convex Functions ◽  
Time Scale ◽  
Integral Inequalities ◽  
New Inequalities

Abstract In this paper, we prove some new integral inequalities for s-convex function on time scale. We give results for the case when time scale is ℝ and when time scale is ℕ.

scholarly journals A Note on Integral Inequalities on Time Scales Associated with Ostrowski’s Type

2019 ◽  
Vol 2019 ◽  
pp. 1-6
Author(s):  
Saeeda Fatima Tahir ◽  
Muhammad Mushtaq ◽  
Muhammad Muddassar
Keyword(s):  
Numerical Integration ◽  
Time Scales ◽  
Time Scale ◽  
Integral Inequalities ◽  
Quantum Calculus

Inequalities become a hot topic for researcher due to its wide applications in means and sum, numerical integration, quantum calculus. Different generalizations and refinements are made by researchers. Here, in this article, we give another generalization of integral inequalities and harmonizing them on time scale T from R.

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