On Refinements of Multidimensional Inequalities of Hardy-Type via Superquadratic and Subquadratic Functions

Zakarya, M.; AlNemer, Ghada; Abd El-Hamid, H. A.; Butush, Roqia; Rezk, H. M.

doi:https://doi.org/10.1155/2022/7668860

Journal of Mathematics

On this page

Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 7668860 | https://doi.org/10.1155/2022/7668860

On Refinements of Multidimensional Inequalities of Hardy-Type via Superquadratic and Subquadratic Functions

M. Zakarya,^1,2Ghada AlNemer ,³H. A. Abd El-Hamid,⁴Roqia Butush,⁵and H. M. Rezk⁶

Academic Editor: Ding-Xuan Zhou

Received29 Jul 2022

Revised15 Sept 2022

Accepted10 Oct 2022

Published24 Nov 2022

Abstract

By utilizing the peculiarities of superquadratic and subquadratic functions, we give the extensions for multidimensional inequalities of Hardy-type with general kernel. We use some algebraic inequalities such as the Minkowski inequality, the refined Jensen inequality, and the Bernoulli inequality to prove the essential results in this paper. The performance of the superquadratic functions is reliable and effective to obtain new dynamic inequalities on time scales. By utilizing special kernels, we also acquire numerous examples and implementations of the related inequalities.

1. Introduction

In [1], Hardy proved that if , over interval and ; then,where the constant is sharp. By rewriting (1) with rather than and taking the limit as , we acquire the limiting case of Hardy’s inequality known as the inequality of Pólya-Knopp ([2]), that is,

In [3], Kaijser et al. specified that both (1) and (2) are special cases of Hardy-Knopp’s inequality.where , is a convex function and is a locally integrable positive function.

In [4], Kaijser et al. applied Fubini’s theorem and Jensen’s inequality to establish an invitingly popularization of (1). Particularly, it was evidenced that if and , such thatand , is a convex function and is defined by

Then, the inequality,holds for any non-negative integrable function such that where is defined by

As a popularization of (6), Krulic et al. [5] have demonstrated that if and are measure spaces with positive -finite measures, and are measurable functions such that is a -integrable function for , is defined asand is defined bywhere . If is a convex function on , then the inequality,holds for any non-negative -integrable function such that where is defined by

In [6], the authors showed that if , , are measurable functions such that is a -integrable function for , is defined by (8) and the function is defined as

Moreover, if is a non-negative superquadratic function, thenholds for any non-negative -integrable function where is defined by (11).

In [7], the researchers demonstrated some Hardy-type inequalities with a general kernel. They have determined that if and are two time scale measure spaces with positive -finite measures, is such that is a convex function and such thatthenholds for all non-negative -integrable such that .

In [8], the authors improved the inequality (16) by replacing the function by an -tuple of functions.such as are -integrable on as follows: let , are non-negative such that is a -integrable function for , and the function are defined by (14), (15), respectively. Then, for a convex function over a convex set , the integral inequality,holds for all -integrable functions such that .

In [9], the authors derived some inequalities of Hardy-type by utilizing the concept of superquadratic functions. Particularly, they proved that if and are measurable functions such that is a -integrable function for , and the function are defined by (14) and (15), respectively. Let be a non-negative superquadratic function. Then, the inequality,holds for all non-negative -integrable function , where is defined by

Moreover, in [10, 11], the authors generalized (19) and proved that if and are measurable functions such that is a -integrable function for , and the function are defined by (14) and (15), respectively. Let be a non-negative superquadratic function. Then, the inequalityholds for all -integrable functions such that , where is defined by

In [12], the authors deduced several generalizations of (19) on time scales. They proved that if , and are measurable functions such that is a -integrable function for , is defined by (14) and the function be defined by

Let be a non-negative superquadratic function. Then, the inequality,holds for all non-negative -integrable function where is defined by (20).

Another development of Hardy-type inequality (24) has been made by Saker et al. [13] as follows: let and be non-negative functions such that is a -integrable function for , is defined byand is defined bywhere . If is a non-negative superquadratic function, then the inequality,holds for all non-negative -integrable function where is defined by

In order to develop dynamic time scale inequalities, we moved the reader to the articles [14–22].

Motivated by the previous results, our major aim in this paper is to deduce several generalizations of general Hardy-type inequalities for multivariate superquadratic functions that involve more general kernels on arbitrary time scales.

The paper is governed as follows: We remember some basic notions, definitions, and results of multivariate superquadratic functions on time scales in Section 2. In Section 3, we prove some new refined dynamic inequalities of Hardy’s type with non-negative kernel by utilizing the peculiarities of superquadratic (or subquadratic) functions. In Section 4, we discuss several particular cases of Hardy-type inequality by choosing such special kernels. Eventually, in Section 5, we give more implementations of our obtained results on particular time scales.

2. Preliminaries

In this section, we will introduce some fundamental concepts and effects to integrals of time scales and for multivariate superquadratic functions which will be useful to deduce our major results.

Before introducing the main results for multidimensional inequalities, it is necessary to present some further definitions. Firstly, suppose , be the Euclidean space. Letbe the function that is defined on . We utilize the following notations:also, for , we write if component wise , , the relations , are defined analogously, is the null vector and . Correspondingly, for , , we define and the n-cells , and are defined similarly. Furthermore, the subsets and in are defined by

In particular,where . Correspondingly, means the set , and .

Now, we arraign the definition and some essential properties of superquadratic functions that are premised in [23].

Definition 1. (see [23]). A function is called superquadratic if for all , there exists a function such thatIf is superquadratic, then is subquadratic and the reverse inequality of (22) is held.

Definition 2. (see [24]). A function is superadditive provided for all . If the reverse inequality holds, then is said to be subadditive.

Lemma 1 (see [25]). Suppose is continuously differentiable and . If is superadditive or is nondecreasing, then is superquadratic.

In the next, we recall a couple of beneficial examples of a superquadratic function.

Example 1. (see [23]). The power function that is defined as is called superquadratic if and subquadratic if (it is also readily seen that if then is a subquadratic function). Since the sum of superquadratic functions is also superquadratic, the function is defined as follows:which is superquadratic on for each (as shown in [23], Example 2) and the function is defined bywhich is superquadratic on for each (see [23], Example 3).

Example 2 (see [23]). Examples 4, 5, and 6 By utilizing the same argument as in Example 4, the functions that defineare superquadratic.
The following lemma shows that non-negative superquadratic functions are indeed convex functions:

Lemma 2 (see [23]). Let be a superquadratic with that is defined as in Definition 1. Then,(i) and ;(ii)If and , then whenever exists for some index at ; where means that the gradient of the function and denotes the partial derivative of over the i-th variable;(iii)If , then is convex and .The following definitions and theorems are referred from [26, 27]. Let , be time scales, andwhich is called an n-dimensional time scale. Let E be -measurable subset of and be a -measurable function. Then, the corresponding -integral is called Lebesgue -integral and is denoted bywhere is a -additive Lebesgue -measure on . Also, ifwhich is an m-tuple of functions in n-variables such that are Lebesgue -integrable on E, then denotes the m-tuple.i.e., -integral acts on each component of .
Particularly, if is an arbitrary time scale and includes only isolated points, thenwhere is referred to as the graininess function on time scale.

Lemma 3 (Minkowski’s inequality [13]). Let and be two finite-dimensional time scale measures spaces. Suppose that , and that are defined on and , respectively. If , thenprovided that all integrals in (23) exist. If ,then (42) is reversed. For , if in addition to (43) andthen (42) is again reversed.
In [25], (Corollary 5.1), Bibi get the following generalization of Jensen’s inequality and the converse of it for superquadratic functions.

Theorem 1 (Jensen’s inequality). Let be a superquadratic function and be -integrable function for and . Then, for every -integrable functions such that and , are -integrable, we have

If is subquadratic, then (45) is reversed.

3. Main Results

In this section, we prove multidimensional Hardy-type inequalities with general kernels on time scales. Before proceeding with results, we introduce the following notations: and are two time scale delta measure spaces with positive -finite measures. be a non-negative so that is -integrable function for and be defined by is -integrable and the function be defined by where .

Theorem 2. Weassume are satisfied. If is non-negative superquadratic function, thenholds for all -integrable functions such that , where is defined byIf and is subquadratic, then (48) is reversed.

Proof. We begin with the following identity:By applying (45) on (50), we findSince and , we haveFurthermore, by employing the Bernoulli inequality [28],It follows that the L. H. S of (52) is not less thanthat is,By multiplying (55) with and integrating it over with respect to , we getApplying (42) on the R. H. S of (56), we getFinally, substituting (57) into (56) and using the definition (47) of the weight function , we haveThis proves (48). The proof of the case in which and is subquadratic is similar; the only difference is that the inequality sign in (48) is reversed. The proof is complete.

Remark 1. For , inequality (48) in Theorem 2 reduces to (27).

Remark 2. For and , inequality (48) in Theorem 2 coincides with (21).

Corollary 1. Given that and are as in Theorem 2 and is measurable function. Since is superquadratic function, then the second term on the L. H. S of (48) is non-negative and the integral inequalityholds.

Remark 3. By taking and , inequality (59) in Corollary 1 reduces to (18).

Remark 4. For and , inequality (59) in Corollary 1 coincides with inequality (2.2) which is [29], (Corollary 2.1.2).

Remark 5. If we rewrite (48) with , or and , then we get

Remark 6. For and , inequality (60) coincides with inequality (3.13) which is [27], (Remark 3.5).

Remark 7. In Remark 5, since , then the second term on the L. H. S. of (60) is non-negative and (60) reduces to the weighted Hardy-type inequality of the formwhich is a refinement of general Hardy-type inequality established in [29] [Remark 2.1.4] and [5].
As a specific case of Theorem 2 when for , we get the next result.

Corollary 2. Suppose that the assumptions of Theorem 2 are satisfied and . Then,If and , then (62) is reversed.

Remark 8. Clearly, for , inequality (62) in Corollary 2 coincides with inequality (46) which is [13], (Corollary 2.1).
In fact, the function is not superquadratic but by working with the superquadratic function (see Lemma 3) and replacing by in Theorem 2, we obtain the next multidimensional version of the Pólya–Knopp type inequality.

Corollary 3. Suppose that the assumptions in Theorem 2 are satisfied and assume thatthenwhereIf , then (64) is reversed.

Remark 9. For , inequality (64) in Corollary 3 coincides with inequality (48) which is [13], (Corollary 2.2).
In the next results, we also suppose the following hypothesis:
Let , , for every , where is an arbitrary time scale and let , .

Theorem 3. Weassume and are satisfied. Suppose such thatwhere . If is non-negative superquadratic function, thenholds for all -integrable functions such that , where is defined byIf is subquadratic function and , then (48) is reversed.

Proof. We get the result from Theorem 2 by taking

Remark 10. For , Theorem 3 coincides with [10], (Corollary 2.11).

Remark 11. In Theorem 3, if we replace by and put , then we get the result given in [13], (Corollary 2.3).

4. Inequalities with Special Kernels

In this section, we get some consequential inequalities of Hardy-type by selecting special kernels.

Theorem 4. Suppose andsuch that is -integrable function and . If is non-negative superquadratic function, thenholds for all -integrable functions such that , where is defined byIf is subquadratic function and , then (71) is reversed.

Proof. We get the result from Theorem 2 by taking
We have the following in this case:, .

Remark 12. For , Theorem 4 coincides with [10], (Corollary 3.1).

Remark 13. For , Theorem 4 coincides with [13], (Corollary 2.7).
If we let , and in Theorem 4, we have the next result.

Corollary 4. Assuming , we definesuch that is -integrable function and . If is non-negative superquadratic function, thenholds for all -integrable functions such that , where is defined byIf is subquadratic function and , then (76) is reversed.

Remark 14. If we take , then inequalities (75) and (76) reduces to

Remark 15. Clearly, for , inequalities (75) and (76), respectively, reduces toandFurthermost, if we take , then (80) becomeswhich is [10], [Remark 3.2].

Theorem 5. Assume with . If is non-negative superquadratic function, thenholds for all -integrable functions such that , where

If is subquadratic function and , then (82) is reversed.

Proof. We get the result from Theorem 2 by takingand .

Corollary 5. Clearly, for inequality (82) reduces towhich is [10], (Corollary 3.3).

5. Inequalities with Particular Time Scales

In this section, we get some consequential inequalities by selecting some specific time scales.

Theorem 6. Assume be an isolated time scale, is -integrable function and , . If , thenholds for all -integrable functions , whereand

Proof. We get the result from Theorem 4 by takingwith , and .

Remark 16. If we take in Theorem 6, then we getwhere , , 2 is defined by (87) andwhich is [10], (Corollary 4.1).

Theorem 7. Assuming be an isolated time scale, is -integrable function and , . Then,holds for all -integrable functions , whereand

Proof. We get the result from Theorem 4 by takingwith and .

Remark 17. If we take in Theorem 7, then we getwhere , is defined by (93) andwhich is [10], (Corollary 4.3).

Theorem 8. Assuming be an isolated time scale, is -integrable function and , . Then,holds for all -integrable functions , whereand

Proof. We get the result from Theorem 4 by takingwith the assumption .

Remark 18. If we take in Theorem 8, then we getwhere , is defined by (99) andwhich is [6], (Corollary 4.4).

Theorem 9. Assuming be an isolated time scale, is -integrable function and , . Then,holds for all -integrable functions , whereand

Proof. We get the result from Theorem 4 by taking

Remark 19. If we take in Theorem 9, then we getwhere , , 2 is defined by (105) andwhich is [10], (Corollary 4.5).

6. Conclusion and Future Work

This research article is dedicated for some general dynamic inequalities of Hardy’s type and their converses on time scales. These inequalities are considered rather in general terms and contain a number of special integral inequalities. In particular, our findings can be seen as refinements of some recent results closely linked to classical Hardy and Pólya–Knopp inequalities on time scale. We use some algebraic inequalities such as the Minkowski inequality, the refined Jensen inequality, and the Bernoulli inequality on time scales to prove the essential results in this paper. Our computed outcomes can be very useful as a starting point to get some continuous inequalities as special cases. In the future, such inequalities can be introduced by using fractional integrals and fractional derivatives of the Riemann−Liouville type on time scales. It will also be very enjoyable to introduce such inequalities in quantum calculus.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

H. M. Rezk, H. A. Abd El-Hamid, R. Butush, and G. AlNemer performed software analysis and wrote the original draft. H. M. Rezk and M. Zakarya reviewed and edited the manuscript. All the authors have read and agreed to the published version of the manuscript.

Acknowledgments

Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2022R45), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

References

G. H. Hardy, “Notes on some points in the integral calculus, LX an inequality between integrals,” Messenger of Math, vol. 54, pp. 150–156, 1925.
View at: Google Scholar
K. Knopp, “Über Reihen mit positiven Gliedern,” Journal of the London Mathematical Society, vol. s1-3, pp. 205–211, 1928.
View at: Publisher Site | Google Scholar
S. Kaijser, L. E. Persson, and A. Öberg, “On carleman and knopp’s inequalities,” Journal of Approximation Theory, vol. 117, pp. 140–151, 2002.
View at: Google Scholar
S. Kaijser, L. Nikolova, L. E. Persson, and A. Wedestig, “Hardy type inequalities via convexity,” Mathematical Inequalities and Applications, vol. 8, no. 3, pp. 403–417, 2005.
View at: Publisher Site | Google Scholar
K. K. H. lreich, J. E. Pečarić, and L. E. Persson, “Some new Hardy type inequalities with general kernels,” Mathematical Inequalities and Applications, vol. 12, no. 3, pp. 473–485, 2009.
View at: Publisher Site | Google Scholar
A. Čižmešija, J. E. Pečarić, and L. E. Persson, “On strengthened Hardy and Pólya-Knopp’s inequalities,” Journal of Approximation Theory, vol. 125, no. 1, pp. 74–84, 2003.
View at: Publisher Site | Google Scholar
M. J. Bohner, A. Nosheen, J. E. Pečarić, and A. Younus, “Some dynamic Hardy-type inequalities with general kernel,” Journal of Mathematical Inequalities, vol. 8, no. 1, pp. 185–199, 2014.
View at: Publisher Site | Google Scholar
T. Donchev, A. Nosheen, and J. E. Pečarić, “Hardy-type inequalities on time scale via convexity in several variables,” ISRN Mathematical Analysis, vol. 2013, Article ID 903196, 9 pages, 2013.
View at: Publisher Site | Google Scholar
J. Adedayo Oguntuase and L. E. Persson, “Time scales Hardy-type inequalities via superquadracity,” Annals of Functional Analysis, vol. 5, no. 2, pp. 61–73, 2014.
View at: Publisher Site | Google Scholar
R. Bibi and W. Ahmad, “Hardy type inequalities for several variables and their converses for time scales integrals via superquadratic functions,” Analysis Mathematica, vol. 45, no. 2, pp. 249–265, 2019.
View at: Publisher Site | Google Scholar
O. O. Fabelurin and J. A. Oguntuase, “Multivariat Hardy-type inequalities on time scales via superquadraticity,” Proceedings of A. Razmadze Mathematical Institute, vol. 167, pp. 29–42, 2015.
View at: Google Scholar
S. H. Saker, H. M. Rezk, and M. Krnić, “More accurate dynamic Hardy-type inequalities obtained via superquadraticity,” Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales Serie A. Matemáticas, vol. 113, no. 3, pp. 2691–2713, 2019.
View at: Publisher Site | Google Scholar
S. H. Saker, H. M. Rezk, I. Abohela, and D. Baleanu, “Refinement multidimensional dynamic inequalities with general kernels and measures,” Journal of Inequalities and Applications, vol. 306, pp. 1–16, 2019.
View at: Google Scholar
S. Abramovich, G. Jameson, and G. Sinnamon, “Refining of Jensen’s inequality,” Bull. Math. Soc. Sci. Math. Roumanie (N.S.), vol. 47, no. 95, pp. 3–14, 2004.
View at: Google Scholar
H. A. A. El-Hamid, H. M. Rezk, A. M. Ahmed, G. AlNemer, M. Zakarya, and H. A. El Saify, “Dynamic inequalities inquotients with general kernels and measures,” Journal of Function Spaces, vol. 2020, Article ID 5417084, 12 pages, 2020.
View at: Publisher Site | Google Scholar
J. Barić, R. Bibi, M. Bohner, and J. Pečarić, “Time scales integral inequalities for superquadratic functions,” Journal of the Korean Mathematical Society, vol. 50, no. 3, pp. 465–477, 2013.
View at: Publisher Site | Google Scholar
H. Feng, S. Hou, L. Y. Wei, and D. X. Zhou, “CNN models for readability of Chinese texts,” Mathematical Foundations of Computing, vol. 5, no. 4, pp. 351–362, 2022.
View at: Publisher Site | Google Scholar
H. Karsli, “On multidimensional Urysohn type generalized sampling operators,” Mathematical Foundations of Computing, vol. 4, no. 4, pp. 271–280, 2021.
View at: Publisher Site | Google Scholar
R. Liu and R. Xu, “Hermite-Hadamard type inequalities for harmonical (h₁, h₂)-convex interval-valued functions,” Mathematical Foundations of Computing, vol. 4, no. 2, pp. 89–103, 2021.
View at: Google Scholar
D. O’Regan, H. M. Rezk, and S. H. Saker, “Some dynamic inequalities involving Hilbert and Hardy-Hilbert operators with kernels,” Results in Mathematics, vol. 73, no. 4, pp. 146–222, 2018.
View at: Publisher Site | Google Scholar
H. M. Rezk, H. A. Abd El-Hamid, A. M. Ahmed, G. AlNemer, and M. Zakarya, “Inequalities of Hardy type via superquadratic functions with general kernels and measures for several variables on time scales,” Journal of Function Spaces, vol. 2020, Article ID 6427378, 15 pages, 2020.
View at: Publisher Site | Google Scholar
D. X. Zhou, “Deep distributed convolutional neural networks: universality,” Analysis and Applications, vol. 16, no. 6, pp. 895–919, 2018.
View at: Publisher Site | Google Scholar
S. Abramovich, S. Banić, and M. Matić, “Superquadratic functions in several variables,” Journal of Mathematical Analysis and Applications, vol. 327, no. 2, pp. 1444–1460, 2007.
View at: Publisher Site | Google Scholar
S. Abramovich, K. Krulic, J. Pečarić, and L.-E. Persson, “Some new refined Hardy type inequalities with general kernels and measures,” Aequationes Mathematicae, vol. 79, no. 1-2, pp. 157–172, 2010.
View at: Publisher Site | Google Scholar
R. Bibi, “Jessen type inequalities for several variables via superquadratic functions,” Banach Journal of Mathematical Analysis, vol. 10, no. 1, pp. 120–132, 2016.
View at: Publisher Site | Google Scholar
M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, MA, USA, 2001.
M. Bohner and A. Peterson, Eds., Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, MA, USA, 2003.
D. S. Mitrinović, J. E. Pečarić, and A. M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Amsterdam, Netherlands, 1993.
K. Rauf and O. A. Sanusi, “Weighted hard-type inequalities on time scales via superquadraticity,” Journal of Mathematical Analysis and Applications, vol. 8, pp. 107–122, 2017.
View at: Google Scholar

Copyright

Copyright © 2022 M. Zakarya et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

215

Downloads

351

Citations