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

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

doi:https://doi.org/10.1155/2020/6427378

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Volume 2020 | Article ID 6427378 | https://doi.org/10.1155/2020/6427378

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

H. M. Rezk,¹H. A. Abd El-Hamid,²A. M. Ahmed,^1,3Ghada AlNemer ,⁴and M. Zakarya^5,6

Academic Editor: Kottakkaran Sooppy Nisar

Received02 Jul 2020

Revised11 Aug 2020

Accepted30 Aug 2020

Published23 Oct 2020

Abstract

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.

1. Introduction

In [1], Hardy claimed this fundamental inequality and proved it: where , and are sharp. They have emerged in the literature since the discovery of (1) numerous papers concerned with new arguments, generalizations, and extensions. One of the most common generalizations for (1) is the disparity of Pólya-Knopp’s inequality (see [2]), which is

In [3], Kaijser et al. signalized that both (1) and (2) are special states of the Hardy-Knopp’s inequality: where is a convex function.

In [4], Cizmeija et al. proved that if is a convex on where with as an integrable function and is defined by then the integral inequality is valid.

In [5], Kaijser et al. applied the inequality of Jensen for convex functions and the theorem of Fubini to establish an invitingly popularization (1). Particularly, it was proved that if and such that and is a convex function, such that be integrable function, and is defined by then the integral inequality is valid, where is defined by

As a popularization of (8), Krulic et al. [6] have demonstrated that if and are two measure spaces with positive finite measures and such that and is a convex function on an interval with be measurable function and is defined by then the integral inequality is valid, where and are defined by

Observe that inequality (12) is a generalization of Hardy inequality (1). Namely, let and , and if and are defined by , then (1) is followed directly from (12), which can be rewritten with instead of and

In the same setting, except with instead of and with relation (12) becomes the Hardy-Hilbert integral inequality (see [7]).

In [8], Abramovich et al. considered a superquadratic function instead of a convex function and obtained the following refinement of inequality (12) in the particular case , as

In [9], Aleksandra et al. proved that, if and are two measure spaces with positive -finite measures, such that is defined as in (10), is a convex function, such that be measurable function and is defined by then the integral inequality is valid, where is defined by (13).

In the past few years, several researchers have suggested the study of dynamic time-scale inequalities. In [10], the authors showed a number of Hardy-type inequalities with a general kernel on time scale. Namely, they have determined that if and are two time-scale measure spaces, such that and such that then the integral inequality is available for all -integrable such that and are a convex function.

Moreover, Donchev et al. [11] improved the inequality (22) by replacing the function by an -tuple of functions such that are -integrable on in the following way. If and are two time-scale measure spaces, a convex set and such that and such that then for every a convex function , the integral inequality is available for all -integrable functions such that

In [12], the authors have specified the time scale version of (17). That is, they proved it if and are two time-scale measure spaces with positive σ-finite measures, and such that is a -integrable function for , and is defined as

If and a superquadratic function, then is available for all -integrable function , and is defined by

In [13], Saker et al. obtained the following refined Jensen’s inequality for superquadratic and in the same paper, he employed the above result to derive the following inequality of Hardy type:

where

, , and such that is a -integrable function for and is defined by (26), is a superquadratic function, and is defined by (29).

Another development of Hardy-type inequality (28) has been made by Bibi [14] and Fabelurin [15] as follows. If and are two time-scale measure spaces, and such that are a -integrable function for is defined by (26) and is a superquadratic function, then is available for all -integrable functions such that , where is defined by

For developing of dynamic inequalities on time scale calculus, we refer the reader to the articles [16–26].

Motivated by the above results, our major aim in this paper is to deduce few nouveau 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 Preliminaries. In Inequalities with General Kernel, we obtain the extensions to the general kernel of Hardy-type inequality. In Inequalities with Specific Time Scales, we extend the latest results from Inequalities with General Kernel to several specific time scales. In Inequalities with Specific Time Scales, we discuss several particular cases of Hardy-type inequality by choosing such special kernels. In Inequalities with Specific Kernels, we derive enhanced forms of certain well-known Hardy-Hilbert-type inequalities.

2. Preliminaries

In this section, we will present some fundamental concepts and effects to integrals of time scales and for multivariate superquadratic functions which will be useful to deduce our major results. Let be the Euclidean space, , , and be the function defined on . Throughout this supplement, we utilize the following notations:

Also, means that , , and is the null vector. The subsets and in are defined by

Now, we arraign the definition and few essential properties of superquadratic functions that premised in [27].

Definition 1. A function is named a superquadratic function if such that If is a superquadratic, then is a subquadratic, and the reverse inequality of (37) is available.

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

Example 1. By [2], Example 1, the power function , defined by , is called a superquadratic if and a subquadratic if (it is also readily seen that if then is a subquadratic function). Since the sum of superquadratic functions is also superquadratic, then is a superquadratic on for each .

Example 2 ([2], Examples 4, 5, and 6,). By utilizing the same argument as in Example 1, the functions defined as are superquadratic.

The following lemma shows that nonnegative superquadratic functions are indeed convex functions.

Lemma 2. Suppose that is a superquadratic with as in Definition 1. Then (i) and (ii)If and , then , whenever exists for some index at (iii)If , then is convex and and .

In the following, we recall the inequality of Minkowski and the inequality of Jensen for superquadratic functions on time scales which are utilized in the proof of the essential results. The following definitions and theorems are referred from [28, 29]. Let be time scales, and is called an -dimensional time scale. Consider to be -measurable subplot of and a -measurable function; then, the corresponding -integral named Lebesgue -integral is denoted by where is a -additive Lebesgue -measure on . Also, if is an -tuple of functions such that are Lebesgue -integrable on , then denotes the -tuple:

i.e., -integral acts on each component of .

Lemma 3. Assume and are two time-scale measure spaces, and suppose that and on and , respectively. If , then is available provided all integrals in (43) exist. If and is available, then (43) is reversed. For , in addition with (44), if is available, then the sign of (43) is reversed.

Theorem 4 ([14], Theorem 3.1). Assume and are two finite-dimensional time-scale measure spaces. Let be continuous and superquadratic, such that is -integrable for . Then, the inequality holds for all functions such that . If is a subquadratic, then (46) is reversed.

3. Inequalities with General Kernel

In this section, we get the Hardy inequality for several variables via multivariate superquadratic functions. Before presenting the results, we labeled the following hypothesis.

(A1) and are two time-scale measure spaces with positive σ-finite measures

(A2) such that

(A3) is -integrable, and the function is defined by where .

Theorem 5. Assume (A1)–(A3) are satisfied. If and is superquadratic, then is available for that is a nonnegative -integrable function such that and defined by If is subquadratic and , then (49) is reversed.

Proof. We begin with an explicit identity By applying the refined Jensen inequality (46) on (51), we find Then, since and , we get Furthermost, by utilizing the famous inequality of Bernoulli, it ensues that the L. H. S. of (53) became that is, we get Multiplying (55) by and integrating it over with respect to , we have Applying the inequality of Minkowski on the R. H. S. of (56), we get Finally, substituting (57) into (56) and utilizing the definition (48) of the weight function , we get which is (49). If is subquadratic and , the corresponding results can be obtained similarly.

Remark 6. If and in Theorem 5, then (49) reduces to (28) premised in Introduction.

Remark 7. For the Lebesgue scale measures and , Theorem 5 coincides with Theorem 2.1.1 in [30].

Remark 8. As a special case of Theorem 5 when and , we have the inequality (19).

Corollary 9. Given that and are as in Theorem 5 and , then, since and superquadratic, the second term on the L. H. S. of (49) is nonnegative and the integral inequality is valid.

Remark 10. By taking in Corollary 9, inequality (59) reduces to (25).

Remark 11. For the Lebesgue scale measures and , Corollary 9 coincides with Corollary 2.1.2 in [30].

Remark 12. Rewrite (49) with such that or ; then

Remark 13. For , inequality (60) coincides with inequality (3.13) in ([28], Remark 3.5).

Remark 14. In Remark 12, since , then the second term on the L. H. S. of (60) is nonnegative. Hence, (60) reduces to which is a refinement of the Hardy-type inequality in ([27], Remark 2.1.4) and [6].

In the following, we labeled some specific superquadratic functions starting with power functions.

Theorem 15. Assume (A1)–(A3) are satisfied. If are -integrable functions such that , then the inequality is valid, where and If and , then (62) is reversed.

Proof. We get the result from Theorem 5 by putting in (49).

Remark 16. For , Theorem 15 reduces to Corollary 3.1 in [13]. In particular, for and , Theorem 15 reduces to Remark 3.11 in [13].

Remark 17. For the Lebesgue scale measures and . Theorem 15 coincides with Corollary 2.1.5 in [30].

Theorem 18. Assume (A1)–(A3) are satisfied. If are -integrable functions such that , then the inequality is valid, where and If , then (65) is reversed.

Proof. We get the result from Theorem 5 by putting in (49) and with instead of .

Remark 19. By taking in Theorem 18, inequality (65) reduces to inequality 3.16 in [28], Corollary 3.2.

Remark 20. For and , the relation (65) that is regarded as a generalization and a refinement of the Pólya-Knopp’s inequality which coincided with Remark 3.12 in [13].

Theorem 21. Assume (A1)–(A3) are satisfied. If are -integrable functions such that , then the inequality is valid, where is defined as in (63). If , then (69) is reversed.

Proof. We get the result from Theorem 5 by putting in (49).

Remark 22. For , Theorem 21 reduces to Theorem 2.5 in [14]. In particular, for and , Theorem 21 coincides with Corollary 2.6 in [14].

Theorem 23. Assume (A1)–(A3) are satisfied. If are -integrable functions such that , then the inequality. is valid, where is defined as in (63). If , then (71) is reversed.

Proof. We get the result from Theorem 5 by putting in (49) with the assumption .

Remark 24. For , Theorem 23 reduces to Theorem 2.7 in [14]. In particular, for and , Theorem 23 coincides with Corollary 2.8 in [14].

Theorem 25. Assume (A1)–(A3) are satisfied. If are -integrable functions such that , then the inequality is valid, where is defined as in (63). If , then (73) is reversed.

Proof. We get the result from Theorem 5 by taking in (49).

Remark 26. For , Theorem 25 reduces to Theorem 2.9 in [14]. In particular, for and , Theorem 25 coincides with Corollary 2.10 in [14].

Now, to wrap up this section, we consider yet another implementation of Theorem 5 rigged with finite measure spaces.

Corollary 27. Let the supposition of Theorem 5 be satisfied and denote and such that : setting and . Then, and Hence, the following inequality is valid. If is subquadratic and , then (76) is reversed.

Remark 28. By taking in Corollary 27, inequality (76) reduces to inequality 3.19 in [28], Corollary 3.2.

Remark 29. For the Lebesgue scale measures and , Corollary 27 coincides with Corollary 2.1.6 in [30].

Remark 30. For , and , Corollary 27 reduces to Corollary 3.3 in [8].

4. Inequalities with Specific Time Scales

In this section, by selecting few different time scales, we get some consequential inequalities. More precisely, assume are points in and . Applying Theorem 5 to , and , we get the following conclusion.

Theorem 31. Assume and such as
Suppose that and where . If and is superquadratic, then is available for all nonnegative integrable functions and for defined as If and are subquadratic, then (78) is reversed.

Remark 32. By taking and replacing, , and , respectively, , and where denotes the characteristic function over in Theorem 31, inequality (78) reduces to inequality 4.1 in [28], Theorem 4.1.

On the other hand, for , consider the set

Then, putting where is a time scale, and . We obtain a dual form of Theorem 31 as follows.

Theorem 33. Suppose that and such that where . If and superquadratic, then is available for all nonnegative -integrable functions and for the operator defined by If is subquadratic and , then (82) is reversed.

Remark 34. By taking and replacing , and , respectively, by , and where denotes the characteristic function over in Theorem 33; inequality (82) reduces to inequality 4.7 in [28], Theorem 4.2.

5. Inequalities with Specific Kernels

In this section, we find some consequential inequalities of the Hardy type by selecting specific kernels and weight functions.

Corollary 35. Suppose that the assumptions of Theorem 31 are satisfied only with Define If and is superquadratic, then (78) is available for all nonnegative -integrable functions defined as If is subquadratic and , then (78) is reversed.

Corollary 36. Assume that the assumptions of Theorem 31 is satisfied only with Define If and is superquadratic, then (78) is available for all nonnegative integrable functions If is subquadratic and , then (78) is reversed.

Corollary 37. Assume that the assumptions of Theorem 31 is satisfied only with defined as and ; then , and in this case is the classical Hardy and denoted by If we let where , then (78) became If is subquadratic and , then (93) is reversed.

Remark 38. For and replacing by and in (93), Corollary 37 coincides with Example 4.1 in [13].

Remark 39. By taking , and replacing by and in (93), we have where If is subquadratic and , then (94) is reversed, which is a refinement of 4.6 in [28], Remark 4.2.

Corollary 40. In Corollary 37, if and , then (93) reduces to where Furthermore, if , then (96) becomes where

Remark 41. For , inequality (96) reduces to where while inequality (98) reduces to

Example 3. Considering Theorem 33 with defined by and , then The operator is defined as and if we let where , then (82) became If is subquadratic and , then (107) is reversed.

Remark 42. For , inequality (107) reduces to where

6. Some Particular Cases

In this section, we obtain a popularization and a refinement of the classical inequality of the Hardy-Hilbert type (16) for numerous variables on time scales. It is clarified in the result below.

Theorem 43. Assume that the assumptions of Theorem 31 are satisfied only with and replace and by the Lebesgue scale measure and .
Furthermore, define If and , then is available for all nonnegative integrable -integrable functions . If , then (111) is reversed.

Proof. Utilizing and in Theorem 15, we obtain and the operator in this case is defined as Utilizing in (62), we obtain Hence, Finally, replacing by in (116), we get (111). The cases and are proved in the same way.

Remark 44. For , Theorem 43 reduces to Theorem 5.1 in [13]. In particular, for , Theorem 43 is a refinement of Theorem 5.5 in [10].

Remark 45. By taking , and , in Theorem 43 and utilizing the known fact that then (111) becomes which is a refinement of (16). For , (118) has been established in [3], Corollary 3.2.

In the following theorem, we introduce a generalized form of (111) on time scales.

Theorem 46. Suppose that and . Furthermore, assume where and ; then is available for all nonnegative integrable functions . If and , then (120) is reversed.

Proof. Rewrite (62) in Theorem 15 with , and . Let us define and We have and the operator in this case is defined as Now, substituting and in (62), we get Hence, Finally, considering (125) with instead of , we obtain (120). The cases and are proved in the same way.

Remark 47. For , Theorem 46 coincides with Theorem 5.2 in [13].

Remark 48. Clearly, for , and , Theorem 46 reduces to Theorem 43.

7. Conclusion and Future Work

The study of dynamic inequalities on time scales has a lot of scope. This research article is devoted to some general Hardy-type dynamic inequalities and their converses on time scales. Inequalities are considered in rather general forms and contain several special integral inequalities. 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. 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. The performance of the superquadratic method for functions is reliable and effective to obtain new dynamic inequalities on time scales. This method has more advantages: it is direct and concise. Thus, the proposed method can be extended to some forms for Hardy’s and related dynamic inequalities in mathematical and physical sciences. Our computed outcomes can be very useful as a starting point to get some continuous inequalities, especially from the obtained dynamic inequalities. In the future, we will get some discrete inequalities from the main results. Also, we will suppose that is an -tuple of functions and is -tuple of variables to get the general forms of Hardy’s and related inequalities on time scales. Similarly, in the future, we can present such inequalities by using Riemann-Liouville-type fractional integrals and fractional derivatives on time scales. It will also be very interesting to present such inequalities on 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

All authors contributed equally. All the authors read and approved the final manuscript.

Acknowledgments

This research was funded by the Deanship of Scientific Research at Princess Nourah Bint Abdulrahman University through the Fast-track Research Funding Program.

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Copyright © 2020 H. M. Rezk 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.

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