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

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.

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Some Opial Dynamic Inequalities Involving Higher Order Derivatives on Time Scales

Discrete Dynamics in Nature and Society ◽

10.1155/2012/157301 ◽

2012 ◽

Vol 2012 ◽

pp. 1-22 ◽

Cited By ~ 4

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.

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Some Dynamic Inequalities via Diamond Integrals for Function of Several Variables

Fractal and Fractional ◽

10.3390/fractalfract5040207 ◽

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.

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Opial and Pólya Type Inequalities Via Convexity

Fasciculi Mathematici ◽

10.1515/fascmath-2018-0009 ◽

2018 ◽

Vol 60 (1) ◽

pp. 145-159 ◽

Cited By ~ 2

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.

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

Symmetry ◽

10.3390/sym12050842 ◽

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.

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

Discrete Dynamics in Nature and Society ◽

10.1155/2012/362526 ◽

2012 ◽

Vol 2012 ◽

pp. 1-23 ◽

Cited By ~ 4

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.

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

Journal of Function Spaces ◽

10.1155/2020/6427378 ◽

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.

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Hardy-Type Inequalities on Time Scale via Convexity in Several Variables

ISRN Mathematical Analysis ◽

10.1155/2013/903196 ◽

2013 ◽

Vol 2013 ◽

pp. 1-9 ◽

Cited By ~ 4

Author(s):

Tzanko Donchev ◽

Ammara Nosheen ◽

Josip Pečarić

Keyword(s):

Time Scales ◽

Convex Functions ◽

Time Scale ◽

Hardy Type Inequalities ◽

Arbitrary Time ◽

Hardy Type ◽

Several Variables ◽

New Inequalities

We extend some Hardy-type inequalities with general kernels to arbitrary time scales using multivariable convex functions. Some classical and new inequalities are deduced seeking applications.

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Inequalities of Hardy type and generalizations on time scales

Analysis ◽

10.1515/anly-2017-0006 ◽

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.

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Some New Inequalities of Opial's Type on Time Scales

Abstract and Applied Analysis ◽

10.1155/2012/683136 ◽

2012 ◽

Vol 2012 ◽

pp. 1-14 ◽

Cited By ~ 4

Author(s):

Samir H. Saker

Keyword(s):

Time Scales ◽

Dynamic Equations ◽

Linear Dynamic ◽

Special Cases ◽

Discrete Inequalities ◽

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 can be applied on the study of distribution of generalized zeros of half-linear dynamic equations on time scales.

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