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 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 Consensus of classical and fractional inequalities having congruity on time scale calculus

2019 ◽  
Vol 27 (1) ◽  
pp. 57-69
Author(s):  
Muhammad Jibril Shahab Sahir
Keyword(s):  
Time Scales ◽  
Time Scale ◽  
Hölder’S Inequality ◽  
Fractional Integrals ◽  
Power Mean ◽  
Extended Form ◽  
Hölder's Inequality ◽  
Time Scale Calculus ◽  
Power Mean Inequality ◽  
Radon’S Inequality

Abstract In this paper, we find accordance of some classical inequalities and fractional dynamic inequalities. We find inequalities such as Radon’s inequality, Bergström’s inequality, Rogers-Hölder’s inequality, Cauchy-Schwarz’s inequality, the weighted power mean inequality and Schlömilch’s inequality in generalized and extended form by using the Riemann-Liouville fractional integrals on time scales.

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 Reconciliation of discrete and continuous versions of some dynamic inequalities synthesized on time scale calculus

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Muhammad Jibril Shahab Sahir
Keyword(s):  
Time Scales ◽  
Time Scale ◽  
Hölder’S Inequality ◽  
Hölder's Inequality ◽  
Time Scale Calculus ◽  
Radon’S Inequality

AbstractThe aim of this paper is to synthesize discrete and continuous versions of some dynamic inequalities such as Radon’s Inequality, Bergström’s Inequality, Schlömilch’s Inequality and Rogers-Hölder’s Inequality on time scales in comprehensive form.

scholarly journals Some fractional dynamic inequalities on time scales of Hardy's type

2020 ◽  
Vol 23 (02) ◽  
pp. 98-109
Author(s):  
A. G. Sayed ◽  
S. H. Saker ◽  
A. M. Ahmed
Keyword(s):  
Time Scales ◽  
Hölder’S Inequality ◽  
Chain Rule ◽  
Integration By Parts ◽  
Hölder's Inequality ◽  
Special Cases

In this paper, we prove some new fractional dynamic inequalities on time scales of Hardy's type due to Yang and Hwang. The results will be proved by employing the chain rule, Hölder's inequality, and integration by parts on fractional time scales. Several well-known dynamic inequalities on time scales will be obtained as special cases from our results.

scholarly journals Some Fractional Dynamic Inequalities of Hardy’s Type via Conformable Calculus

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 434 ◽  
Author(s):  
Samir Saker ◽  
Mohammed Kenawy ◽  
Ghada AlNemer ◽  
Mohammed Zakarya
Keyword(s):  
Time Scales ◽  
Time Scale ◽  
Hölder’S Inequality ◽  
Chain Rule ◽  
Hölder's Inequality ◽  
Conformable Calculus

In this article, we prove some new fractional dynamic inequalities on time scales via conformable calculus. By using chain rule and Hölder’s inequality on timescales we establish the main results. When α = 1 we obtain some well-known time-scale inequalities due to Hardy, Copson, Bennett and Leindler inequalities.

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 Hilbert-type inequalities for time scale nabla calculus

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
H. M. Rezk ◽  
Ghada AlNemer ◽  
H. A. Abd El-Hamid ◽  
Abdel-Haleem Abdel-Aty ◽  
Kottakkaran Sooppy Nisar ◽  
...  
Keyword(s):  
Time Scale ◽  
Hölder’S Inequality ◽  
Hölder's Inequality ◽  
Arbitrary Time ◽  
Hilbert Type

Abstract This paper deals with the derivation of some new dynamic Hilbert-type inequalities in time scale nabla calculus. In proving the results, the basic idea is to use some algebraic inequalities, Hölder’s inequality, and Jensen’s time scale inequality. This generalization allows us not only to unify all the related results that exist in the literature on an arbitrary time scale, but also to obtain new outcomes that are analytical to the results of the delta time scale calculation.

Mathematical Morphology and Fourier Analysis on Time Scales

2009 ◽  
Author(s):  
Robert J Marks II
Keyword(s):  
Image Processing ◽  
Fourier Analysis ◽  
Mathematical Morphology ◽  
Time Scales ◽  
Time Scale ◽  
Time Signal ◽  
Scale Theory ◽  
Black And White ◽  
Introductory Material ◽  
Special Cases

Mathematical morphology, used extensively in image processing, tracks the support domains for the operation of convolution and deconvolution. Morphology is also important in the modelling of signals on time scales. Time scale theory provides a generalization tent under which the operations of discrete and continuous time signal and Fourier analysis rest as special cases. The time scale paradigm provides modelling under which a rich class of hybrid signals and systems can be analyzed. We begin with introductory material on mathematical morphology which is foundational to the development of time scale theory. The support of convolution is related to the operation of dilation in mathematical morphology. Mathematical morphology is most commonly associated with image processing. Applications of morphology was initially applied to binary black and white images by Matheron [966]. The field is richly developed [506, 578]. Here, we outline the fundamentals. In N dimensions, let X and H denote a set of vectors or, in the degenerate case of one dimension, a set of real numbers.

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.

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