scholarly journals Weighted dynamic inequalities of Opial-type on time scales

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
A. A. El-Deeb ◽  
Fatma M. Kh ◽  
Gamal A. F. Ismail ◽  
Zareen A. Khan
Keyword(s):  
Time Scales ◽  
Hölder Inequality ◽  
Chain Rule ◽  
Simple Consequence ◽  
Special Cases ◽  
Discrete Inequalities

Abstract In this paper, we will state and prove some weighted dynamic inequalities of Opial-type involving integrals of powers of a function and of its derivative on time scales which not only extend some results in the literature but also improve some of them. The main results will be proved by using some algebraic inequalities, the Hölder inequality and a simple consequence of Keller’s chain rule on time scales. As special cases of the obtained dynamic inequalities, we will get some continuous and discrete inequalities.

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 Some Opial Dynamic Inequalities Involving Higher Order Derivatives on Time Scales

2012 ◽  
Vol 2012 ◽  
pp. 1-22 ◽  
Author(s):  
Samir H. Saker
Keyword(s):  
Time Scales ◽  
Hölder’S Inequality ◽  
Higher Order ◽  
Chain Rule ◽  
Simple Consequence ◽  
Hölder's Inequality ◽  
Special Cases ◽  
Discrete Inequalities

We will prove some new Opial dynamic inequalities involving higher order derivatives on time scales. The results will be proved by making use of Hölder's inequality, a simple consequence of Keller's chain rule and Taylor monomials on time scales. Some continuous and discrete inequalities will be derived from our results as special cases.

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 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 On Hilbert's inequality on time scales

2017 ◽  
Vol 11 (2) ◽  
pp. 399-423 ◽  
Author(s):  
Saker Saker ◽  
A.A. El-Deeb ◽  
H.M. Rezk ◽  
Ravi Agarwal
Keyword(s):  
Time Scales ◽  
Jensen’S Inequality ◽  
Chain Rule ◽  
Simple Consequence ◽  
Jensen's Inequality ◽  
Hilbert’S Inequality ◽  
Special Cases

In this paper, we will prove some new dynamic inequalities of Hilbert's type on time scales. Our results as special cases extend some obtained dynamic inequalities on time scales.and also contain some integral and discrete in- equalities as special cases. We prove our main results by using some algebraic inequalities, H?older's inequality, Jensen's inequality and a simple consequence of Keller's chain rule on time scales.

scholarly journals New Inequalities of Opial's Type on Time Scales and Some of Their Applications

2012 ◽  
Vol 2012 ◽  
pp. 1-23 ◽  
Author(s):  
Samir H. Saker
Keyword(s):  
Time Scales ◽  
Second Order ◽  
Dynamic Equations ◽  
Linear Dynamic ◽  
Special Cases ◽  
Discrete Inequalities ◽  
Zeros Of Solutions ◽  
Yield Conditions ◽  
New Inequalities

We will prove some new dynamic inequalities of Opial's type on time scales. The results not only extend some results in the literature but also improve some of them. Some continuous and discrete inequalities are derived from the main results as special cases. The results will be applied on second-order half-linear dynamic equations on time scales to prove several results related to the spacing between consecutive zeros of solutions and the spacing between zeros of a solution and/or its derivative. The results also yield conditions for disfocality of these equations.

scholarly journals New Characterizations of Weights in Hardy and Opial Type Inequalities via Solvability of Dynamic Equations

2019 ◽  
Vol 2019 ◽  
pp. 1-13
Author(s):  
S. H. Saker ◽  
A. G. Sayed ◽  
A. Sikorska-Nowak ◽  
I. Abohela
Keyword(s):  
Time Scales ◽  
Second Order ◽  
Dynamic Equations ◽  
Special Cases ◽  
Discrete Inequalities

In this paper, we prove that the solvability of dynamic equations of second order is sufficient for the validity of some Hardy and Opial type inequalities with two different weights on time scales. In particular, the results give new characterizations of two different weights in inequalities containing Hardy and Opial operators. The main contribution in this paper is the characterizations of weights in discrete inequalities that will be formulated from our results as special cases.

SOME REVERSE DYNAMIC INEQUALITIES ON TIME SCALES

2017 ◽  
Vol 96 (3) ◽  
pp. 445-454 ◽  
Author(s):  
R. P. AGARWAL ◽  
R. R. MAHMOUD ◽  
D. O’REGAN ◽  
S. H. SAKER
Keyword(s):  
Time Scales ◽  
Hölder Inequality ◽  
Chain Rule ◽  
Fubini Theorem ◽  
Reverse Hölder Inequality ◽  
Reverse Hölder

In this paper, we prove some new reverse dynamic inequalities of Renaud- and Bennett-type on time scales. The results are established using the time scales Fubini theorem, the reverse Hölder inequality and a time scales chain rule.

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 Some Opial-Type Inequalities on Time Scales

2011 ◽  
Vol 2011 ◽  
pp. 1-19 ◽  
Author(s):  
S. H. Saker
Keyword(s):  
Time Scales ◽  
Special Cases ◽  
Discrete Inequalities

We will prove some dynamic inequalities of Opial type on time scales which not only extend some results in the literature but also improve some of them. Some discrete inequalities are derived from the main results as special cases.

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