Inequalities of Hardy type and generalizations on time scales

Analysis ◽  
2018 ◽  
Vol 38 (1) ◽  
pp. 47-62
Author(s):  
Samir H. Saker ◽  
Mahmoud M. Osman ◽  
Donal O’Regan ◽  
Ravi P. Agarwal
Keyword(s):  
Time Scales ◽  
Hardy Type ◽  
Special Cases ◽  
Discrete Inequalities

AbstractIn this paper, we prove some new dynamic inequalities on time scales which as special cases contain several generalizations of integral and discrete inequalities due to Hardy, Copson, Leindler, Bennett, Pachpatte and Pečarić and Hanjš.

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 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.

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 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 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 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.

A critical oscillation constant as a variable of time scales for half-linear dynamic equations

2010 ◽  
Vol 60 (2) ◽  
Author(s):  
Pavel Řehák
Keyword(s):  
Time Scales ◽  
Type Inequality ◽  
Dynamic Equations ◽  
Hardy Type Inequality ◽  
Linear Dynamic ◽  
Hardy Type ◽  
Special Cases ◽  
Critical Oscillation ◽  
Critical Constant ◽  
Oscillation Constant

AbstractWe present criteria of Hille-Nehari type for the half-linear dynamic equation (r(t)Φ(y Δ))Δ+p(t)Φ(y σ) = 0 on time scales. As a particular important case we get that there is a a (sharp) critical constant which may be different from what is known from the continuous case, and its value depends on the graininess of a time scale and on the coefficient r. As applications we state criteria for strong (non)oscillation, examine generalized Euler type equations, and establish criteria of Kneser type. Examples from q-calculus, a Hardy type inequality with weights, and further possibilities for study are presented as well. Our results unify and extend many existing results from special cases, and are new even in the well-studied discrete 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 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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