scholarly journals Some Dynamic Hilbert-Type Inequalities on Time Scales

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1410 ◽  
Author(s):  
Ghada AlNemer ◽  
Mohammed Zakarya ◽  
Hoda A. Abd El-Hamid ◽  
Praveen Agarwal ◽  
Haytham M. Rezk
Keyword(s):  
Time Scales ◽  
Time Scale ◽  
Discrete Inequalities ◽  
Hilbert Type ◽  
General Time

Throughout this article, we will demonstrate some new generalizations of dynamic Hilbert type inequalities, which are used in various problems involving symmetry. We develop a number of those symmetric inequalities to a general time scale. From these inequalities, as particular cases, we formulate some integral and discrete inequalities that have been demonstrated in the literature and also extend some of the dynamic inequalities that have been achieved in time scales.

scholarly journals New Weighted Opial-Type Inequalities on Time Scales for Convex Functions

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 842
Author(s):  
Ahmed A. El-Deeb ◽  
Dumitru Baleanu
Keyword(s):  
Time Scales ◽  
Convex Functions ◽  
Time Scale ◽  
Hölder Inequality ◽  
Special Cases ◽  
Discrete Inequalities ◽  
General Time

Our work is based on the multiple inequalities illustrated in 1967 by E. K. Godunova and V. I. Levin, in 1990 by Hwang and Yang and in 1993 by B. G. Pachpatte. With the help of the dynamic Jensen and Hölder inequality, we generalize a number of those inequalities to a general time scale. In addition to these generalizations, some integral and discrete inequalities will be obtained as special cases of our results.

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 dynamic Hilbert-type inequalities for two variables on time scales

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
H. A. Abd El-Hamid ◽  
H. M. Rezk ◽  
A. M. Ahmed ◽  
Ghada AlNemer ◽  
M. Zakarya ◽  
...  
Keyword(s):  
Time Scales ◽  
Special Cases ◽  
Discrete Inequalities ◽  
Hilbert Type

AbstractIn this paper, we discuss some new Hilbert-type dynamic inequalities on time scales in two separate variables. We also deduce special cases, like some integral and their respective discrete inequalities.

scholarly journals More on Hölder’s Inequality and It’s Reverse via the Diamond-Alpha Integral

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1716
Author(s):  
M. Zakarya ◽  
H. A. Abd El-Hamid ◽  
Ghada AlNemer ◽  
H. M. Rezk
Keyword(s):  
Linear Combination ◽  
Time Scales ◽  
Time Scale ◽  
Hölder’S Inequality ◽  
Hölder's Inequality ◽  
Special Cases ◽  
General Time

In this paper, we investigate some new generalizations and refinements for Hölder’s inequality and it’s reverse on time scales through the diamond-α dynamic integral, which is defined as a linear combination of the delta and nabla integrals, which are used in various problems involving symmetry. We develop a number of those symmetric inequalities to a general time scale. Our results as special cases extend some integral dynamic inequalities and Qi’s inequalities achieved on time scales and also include some integral disparities as particular cases when T=R.

scholarly journals Some Dynamic Inequalities of Hilbert’s Type

2020 ◽  
Vol 2020 ◽  
pp. 1-13 ◽  
Author(s):  
A. M. Ahmed ◽  
Ghada AlNemer ◽  
M. Zakarya ◽  
H. M. Rezk
Keyword(s):  
Time Scales ◽  
Hölder’S Inequality ◽  
Jensen’S Inequality ◽  
Jensen's Inequality ◽  
Hölder's Inequality ◽  
Discrete Inequalities ◽  
Special Case ◽  
Hilbert Type

This paper is concerned with deriving some new dynamic Hilbert-type inequalities on time scales. The basic idea in proving the results is using some algebraic inequalities, Hölder’s inequality and Jensen’s inequality, on time scales. As a special case of our results, we will obtain some integrals and their corresponding discrete inequalities of Hilbert’s type.

Extensions of Dynamic Inequalities of Hardy’s Type on Time Scales

2015 ◽  
Vol 65 (5) ◽  
Author(s):  
S. H. Saker ◽  
Donal O’Regan
Keyword(s):  
Time Scales ◽  
Time Scale ◽  
Hölder Inequality ◽  
Chain Rule ◽  
Simple Consequence ◽  
Hardy Type ◽  
Special Cases ◽  
Discrete Inequalities ◽  
New Inequalities

AbstractIn this paper using some algebraic inequalities, Hölder inequality and a simple consequence of Keller’s chain rule we prove some new inequalities of Hardy type on a time scale T. These inequalities as special cases contain some integral and discrete inequalities when T = ℝ and T = ℕ.

scholarly journals Opial and Pólya Type Inequalities Via Convexity

2018 ◽  
Vol 60 (1) ◽  
pp. 145-159 ◽  
Author(s):  
S. H. Saker ◽  
D. M. Abdou ◽  
I. Kubiaczyk
Keyword(s):  
Time Scales ◽  
Time Scale ◽  
Hölder’S Inequality ◽  
Jensen’S Inequality ◽  
Chain Rule ◽  
Integral Inequalities ◽  
Taylor Formula ◽  
Special Cases ◽  
Discrete Inequalities ◽  
Classical Integral

Abstract In this paper, we prove some new dynamic inequalities related to Opial and Pólya type inequalities on a time scale 𝕋. We will derive the integral and discrete inequalities of Pólya’s type as special cases and also derive several classical integral inequalities of Opial’s type that has been obtained in the literature as special cases. The main results will be proved by using the chain rule, Hölder’s inequality and Jensen’s inequality, Taylor formula on time scales.

scholarly journals Dynamic Hilbert-Type Inequalities with Fenchel-Legendre Transform

Symmetry ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 582 ◽  
Author(s):  
Ahmed A. El-Deeb ◽  
Samer D. Makharesh ◽  
Dumitru Baleanu
Keyword(s):  
Time Scale ◽  
Legendre Transform ◽  
Special Cases ◽  
Hilbert Type ◽  
General Time

Our work is based on the multiple inequalities illustrated in 2020 by Hamiaz and Abuelela. With the help of a Fenchel-Legendre transform, which is used in various problems involving symmetry, we generalize a number of those inequalities to a general time scale. Besides that, in order to get new results as special cases, we will extend our results to continuous and discrete calculus.

scholarly journals Itô’s Formula, the Stochastic Exponential, and Change of Measure on General Time Scales

2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
Wenqing Hu
Keyword(s):  
Time Scales ◽  
Time Scale ◽  
Closed Form Expression ◽  
Itô’S Formula ◽  
Change Of Measure ◽  
Quantum Time ◽  
Itô's Formula ◽  
Ito’S Formula ◽  
General Time ◽  
Ito's Formula

We provide an Itô formula for stochastic dynamical equation on general time scales. Based on this Itô’s formula we give a closed-form expression for stochastic exponential on general time scales. We then demonstrate Girsanov’s change of measure formula in the case of general time scales. Our result is being applied to a Brownian motion on the quantum time scale (q-time scale).

scholarly journals Almost Automorphic Functions on the Quantum Time Scale and Applications

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Yongkun Li
Keyword(s):  
Time Scales ◽  
Time Scale ◽  
Dynamic Equations ◽  
Almost Periodic ◽  
Almost Automorphic ◽  
Almost Automorphic Functions ◽  
Quantum Time ◽  
Almost Periodic Time Scales ◽  
Automorphic Functions ◽  
General Time

We first propose two types of concepts of almost automorphic functions on the quantum time scale. Secondly, we study some basic properties of almost automorphic functions on the quantum time scale. Then, we introduce a transformation between functions defined on the quantum time scale and functions defined on the set of generalized integer numbers; by using this transformation we give equivalent definitions of almost automorphic functions on the quantum time scale; following the idea of the transformation, we also give a concept of almost automorphic functions on more general time scales that can unify the concepts of almost automorphic functions on almost periodic time scales and on the quantum time scale. Finally, as an application of our results, we establish the existence of almost automorphic solutions of linear and semilinear dynamic equations on the quantum time scale.

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