Hardy-Leindler-Type Inequalities via Conformable Delta Fractional Calculus

Abstract

In this article, some fractional Hardy-Leindler-type inequalities will be illustrated by utilizing the chain law, Hölder’s inequality, and integration by parts on fractional time scales. As a result of this, some classical integral inequalities will be obtained. Also, we would have a variety of well-known dynamic inequalities as special cases from our outcomes when .

1. Introduction

The Hardy discrete inequality is known as (see [1]) where for all .

In [2], Hardy employed the calculus of variations and exemplified the continuous version for (1) as follows: where is integrable over any finite interval is convergent and integrable over , and is a sharp constant in (1) and (2).

Leindler in [3] exemplified that if and then

The converses of (3) and (4) are exemplified by Leindler in [4]. Precisely, he established that if then

Saker [5] exemplified the time scale version of (3) and (4), respectively, as follows: suppose that be a time scale and If , for any then

Also, if and , for any then

The converses of (7) and (8) are established by Saker [5]. Precisely, he exemplified that, if is a time scale, , and , for any then

Also, if and , for any then which are the time scale version for (5) and (6), respectively. For developing dynamic inequalities, see the papers ([6–11]).

Our target in this article is proving some fractional dynamic inequalities for Hardy-Leindler’s type, and it is reversed with employing conformable calculus on time scales. This article is structured as follows: In Section 2, we discuss the preliminaries of conformable fractional on time scale calculus which will be required in proving our main outcomes. In Section 3, we will exemplify the major consequences.

2. Basic Concepts

In this part, we introduce the essentials of conformable fractional integral and derivative of order on time scales that will be used in this article (see [12–15]). For a time scale , we define the operator , as

Also, we define the function by

Finally, for any we refer to the notation by i.e., . In the following, we define conformable -fractional derivative and -fractional integral on .

Definition 1 (see [16], Definition 3.1). Suppose that and Then, for , we define to be the number with the property that, for any there is a neighborhood of s.t. we have The conformable -fractional derivative on at is

Theorem 2 (see [16], Theorem 3.6). Assume and are conformable -fractional derivatives at Then, we have the following. (i)The sum is a conformable -fractional derivative and (ii)The product is a conformable -fractional derivative with (iii)If , then is a conformable -fractional derivative with

Lemma 3 (Chain rule). Suppose that is continuous and -fractional differentiable at , for and is continuously differentiable. Then, is -fractional differentiable and

Definition 4 (see [16], Definition 4.1). For then the -conformable fractional integral of is defined as

Theorem 5 (see [16], Theorem 4.3). Let , and be rd-continuous functions. Then, (i)(ii)(iii)(iv)(v)

Lemma 6 (Integration by parts formula [16], Theorem 4.3). Suppose that where If are rd-continuous functions and , then

Lemma 7 (Hölder’s inequality). Let where If and , then where and
Through our paper, we will consider the integrals are given exist (are finite, i.e., convergent).

3. Main Results

Here, we will exemplify our major results in this article. In the pursuing theorem, we will exemplify Leindler’s inequality (7) for fractional time scales as follows.

Theorem 8. Suppose that be a time scale and If and , for any then

Proof. By utilizing (20) on with and we have where Substituting (26) into (25), we get Using and in (27), we have Utilizing the chain rule (18), we get Since , we get Substituting (30) into (28) yields Inequality (31) can be written as Implementing Hölder’s inequality on the R.H.S of (32) with indices , we get By substituting (33) into (32), we get which is (23).

Corollary 9. At in Theorem 8, then where , and , for any

Remark 10. In Corollary 9, if we divide both sides of (35) by the factor and using the fact that then Elevating the last inequality to the th power, we get which is (7) in Introduction.

Remark 11. If we put (i.e., ) in Theorem 5, then where , and for any

Remark 12. Clearly, for and Remark 12 coincides with Remark 10 in [5].

Remark 13. When (i.e., ) , and in (23), then we get If , then (40) becomes which is Remark 11 in [5].

In the pursuing theorem, we will exemplify Leindler’s inequality (8) on fractional time scales as follows.

Theorem 14. Suppose that be a time scale and If , and , for any then

Proof. By utilizing (20) on with and we have where Substituting (45) into (44), we get Using the fact that and (46) became Utilizing chain rule (18), we get By substituting (48) into (47), we get Inequality (49) can be written as Implementing Hölder’s inequality on the R.H.S of (50) with indices , we get By substituting (51) into (50), we get which is (42).

Corollary 15. At in Theorem 14, then where , and , for any

Remark 16. In Corollary 15, if we divide both sides of (53) by the factor and using the fact that then Elevating the last inequality to the th power, we get which is (8) in Introduction.

Remark 17. As a result, if (i.e., ) in Theorem 14, then where , and , for any

Remark 18. Clearly, for and Remark 17 coincides with Remark 12 in [5].

Remark 19. When (i.e., ), , and in (42), we get If , then (58) becomes which is Remark 13 in [5].

In the pursuing theorem, we will exemplify Leindler’s inequality (9) for fractional time scales as follows.

Theorem 20. Suppose that be a time scale and If and , for any then

Proof. By applying (20) on with and we have where Substituting (63) into (62) yields Using the fact that and (64) became Utilizing chain rule (18), we get Since , we obtain By substituting (67) into (65), we have Raises (68) to the factor we get By applying Hölder’s inequality on with indices , and we see that This implies that By substituting (73) into (69), we get which is (60).

Corollary 21. At in Theorem 20, then where , and , for any

Remark 22. In Corollary 21, if we divide both sides of (75) by the factor then (75) can be written as which is (9) in Introduction.

Remark 23. As a result, if (i.e., ) in Theorem 20, then where , and , for any

Remark 24. Clearly, for and Remark 23 coincides with Remark 16 in [5].

Remark 25. When (i.e., ), , and in (60), then we get If , then (79) becomes which is Remark 17 in [5].

In the pursuing theorem, we will exemplify Leindler’s inequality (10) for fractional time scales as follows.

Theorem 26. Suppose that be a time scale and If , and , for any then

Proof. By applying (20) on with and we have where Substituting (84) into (83), we get Using the fact that and (85) became Utilizing chain rule (18), we have Since , we get By substituting (88) into (86), we get Raises (89) to the factor we have By applying Hölder’s inequality on with indices , and we see that This implies that By substituting (94) into (90), we get which is (81).

Corollary 27. At in Theorem 26, then where , and , for any

Remark 28. In Corollary 27, if we divide both sides of (96) by the factor then (96) can be written as which is (10) in Introduction.

Remark 29. As a result, if (i.e., ) in Theorem 26, then where , and , for any

Remark 30. Clearly, for and , Remark 29 coincides with Remark 18 in [5].

Remark 31. When (i.e., ), , and in (81), then we get If , then (100) becomes which is Remark 19 in [5].

4. Conclusions and Future Work

In this article, we explore new generalizations of the integral Hardy-Leindler-type inequalities by the utilization of the delta conformable calculus on time scales which are used in various problems involving symmetry. We generalize a number of those inequalities to a general time scale measure space. In addition to this, in order to obtain some new inequalities as special cases, we also extend our inequalities to a discrete and continuous calculus. In future work, we will continue to generalize more fractional dynamic inequalities by using Specht’s ratio, Kantorovich’s ratio, and n-tuple fractional integral.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this project under grant number (R.G.P. 2/29/43). The authors are thankful to Taif University and Taif University researchers supporting project number (TURSP-2020/160), Taif University, Taif, Saudi Arabia.