`ISRN Mathematical AnalysisVolume 2013 (2013), Article ID 903196, 9 pageshttp://dx.doi.org/10.1155/2013/903196`
Research Article

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

1Department of Mathematics, University Al. I. Cuza, 700506 Iaşi, Romania
2Abdus Salam School of Mathematical Sciences, Government College University, Lahore 54600, Pakistan
3University of Zagreb, Faculty of Textile Technology, 10000 Zagreb, Croatia

Received 3 June 2013; Accepted 1 July 2013

Academic Editors: M. Lindstrom and S. Liu

Copyright © 2013 Tzanko Donchev 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.

Abstract

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.

1. Introduction

The significant Hardy inequality is published in [1] (1952) (both in the continuous and discrete settings). More general Hardy integral inequalities have been studied in continuous cases. We notice only [26] and the references therein.

Recently, this inequality is studied in discrete case, and some variants of it are proved in case of time scales [79].

In [10], the authors study Hardy-type inequalities using convex functions of one variable with general kernels to arbitrary time scales.

The aim of this paper is to provide Hardy-type inequalities using multivariable convex function with general kernels to arbitrary time scales. Notice that time scales include continuous and discrete time cases under unified approach.

Firstly, we recall necessary preliminary facts needed afterward. The main results are given in Section 3. Section 4 is devoted to some inequalities with certain kernels. In the last section, we discuss some particular cases of Hardy-type inequalities.

2. Preliminaries

First, we recall the basic concepts used in the paper and refer the interested reader to [11] for the theory of time scales. A time scale is any nonempty closed subset of the real line . On nondensity points we define forward, respectively, backward jump operators as The point is said to be right-scattered if and left-scattered if , respectively. Clearly, is right-dense if and left dense if , respectively.

Let ; we define -dimensional time scale by the Cartesian product of given time scales , , as Evidently, is a complete metric space with distance as follows: Now we are going to describe the construction of Lebesgue measure in . We refer to [1214] for the theory of measure spaces and measurable functions on time scales.

Let be the family of all -dimensional time scale intervals in ; that is, with , , and for all . Let be the set function that assigns to each -dimensional time scale interval its volume as follows: Let . If there exists finite or countable system of pairwise disjoint -dimensional time scale intervals with , then the outer measure of is defined by If there is no such covering of , then .

A subset of is said to be measurable (or -measurable) if holds for all , where . The family of all -measurable subsets of is a -algebra generated by . The restriction of to , which we denote by , is a -additive measure on . Clearly, and for each . The measure (called the Lebesgue -measure on ) is Carathéodory extension of the original measure defined on . We call an -dimensional time scale measure space.

Denote . Similarly, . Let be the set of all points for which there exists at least one such that . From [11, Theorem 3.1], we know that if , then the single-point set is -measurable and Obviously, for all , the set can be represented as a finite or countable union of intervals of the family ; hence it is -measurable. Furthermore, the set is -measurable being the difference of two -measurable sets and , but does not have a finite or countable covering intervals of , that is, for any -measurable subset of such that has -measure . In particular, if , where is an arbitrary time scale, then the set is -measurable.

The function is said to be -measurable if for every , the set is -measurable. It is easy to see that is -measurable if and only if for each open set , the set is -measurable. Moreover, if is -measurable and is a continuous function, then is -measurable.

Having the -additive measure on , we possess the corresponding integration theory for functions , according to the general Lebesgue integration theory (see, e.g., [15]). The Lebesgue integral associated with the measure on is called the Lebesgue -integral. For a -measurable set and a -measurable function , the corresponding -integral of over will be denoted by Notice that all theorems of the general Lebesgue integration theory, including the Lebesgue dominated convergence theorem, hold also for Lebesgue -integrals on .

If is a time scale and the interval contains only isolated points, then

Let and be two finite dimensional time scale measure spaces. We define the product measure space , where is the product -algebra generated by and Fubini's Theorem in Time Scales see ([16, Theorem 1.1]). If is a -integrable function and if we define the function for a.e. and for a.e. , then is -integrable on , is -integrable on , and

3. Inequalities with General Kernels

Let be -tuple of functions such that are -integrable for all . Then denotes the -tuple ; that is, -integral acts on each component of .

The following Jensen's inequality on time scales is given in [17, Theorem 2.1].

Theorem 1. Let and be two time scale measure spaces. Suppose is a closed convex set, is convex, and . Moreover, let be nonnegative such that is -integrable. Then one has

Throughout this section, we assume that the following hypotheses hold.H1 and are two time scale measure spaces.H2 is such that .H3 is such that .

Theorem 2. If is a closed convex set such that is convex and continuous, then holds for all -integrable functions such that .

Proof. Using Jensen's inequality (13) for several variables and the Fubini theorem on time scales, we find that The proof is therefore complete.

Remark 3. If is concave, then (14) holds in reverse direction.

Corollary 4. Let be continuous function and define for all . If is convex, then holds for all and continuous monotone functions such that are -integrable for all .

Proof. Replace in Theorem 2   with for all and with .

Remark 5. In case that and , Corollary 4 is as [18, Corollary 1.2].

Remark 6. In case that , we can use the results of Beck [19] (see also [20], page 194) as applications of Corollary 4, which corresponds to the generalizations of Hölder’s and Minkoski's inequalities. In classical case, many authors have studied these types of generalizations; see, for example, [2123].

Furthermore, in the paper, we use with .

Corollary 7. If , then holds for all -integrable , where .

Proof. Use in Corollary 4; then is concave in Theorem 2.

Corollary 8. If , then holds for all -integrable , where .

Proof. Use , , , in Corollary 4; then is concave in Theorem 2.

Remark 9. If , then reverse inequalities hold in Corollaries 4, 7, and 8.

4. Inequalities with Special Kernels

Throughout this section and in the next section, we assume that the following hypothesis holds.H4, for all , where is an arbitrary time scale.

Corollary 10. Assume that If is a closed convex set such that is convex and continuous; then holds for all -integrable such that , where

Proof. Statement follows from Theorem 2 using since in this case

Corollary 11. Assume that If is a closed convex set such that is convex and continuous, then (21) holds for all -integrable such that , and

Proof. Statement follows from Theorem 2 using since in this case

Theorem 12. Assume that If is a closed convex set such that is convex and continuous, then holds for all -integrable such that , where

Proof. Statement follows from Theorem 2 using since in this case Thus .

Corollary 13. If for all in and is a closed convex set such that is convex and continuous, then holds for all -integrable such that , where

Proof. The statement follows from Theorem 12 using , since in this case

Example 14. If for all in addition to the assumptions of Corollary 13, then (34) takes the form

Remark 15. Clearly, if the left-hand side is in (34), then right-hand side is also .

Remark 16. For and , inequality (37) is proved in [4, 24].

5. Some Particular Cases

In this section, firstly, we give Hilbert-type inequality on time scales.

Theorem 17. If , , in and for one defines then holds for all -integrable , where .

Proof. Use and in Corollary 7 to obtain where Using this value in (40), we obtain Now, if we replace with and with , we obtain (39).

Another Proof of (42). Consider L.H.S of (40) and applying Hölder inequality on time scale [25, Theorem 6.2] and Fubini theorem on time scale [16, Theorem 1.1], we have

Example 18. It is known that for all with . If ; then from (39), we obtain
In the rest of paper, we take , , in .

Theorem 19. If (38) is satisfied, then holds for all -integrable , where .

Proof. Use and in Corollary 7 to obtain Now, if we replace with and with , we obtain (46).

Remark 20. (a) We can give another proof of (47) using Minkowski's inequality on time scale [25, Theorem  7.2].
(b) If , then we have reverse inequalities.
Now we consider some generalizations of the Pólya-Knopp type inequalities.

Corollary 21. If (29) holds, and furthermore, if is a closed convex set such that is convex and continuous, then holds for all -integrable such that .

Proof. The statement follows from Theorem 12 using .

Corollary 22. Assume (29). Then holds for all -integrable , where .

Proof. Statement follows from Corollary 7 using .

Corollary 23. Assume (29). Then holds for all -integrable , where .

Proof. Statement follows from Corollary 8 using .

Example 24. When consists of isolated points, then from Corollary 22, we have where .

Example 25. When consists of isolated points, then from Corollary 23, we have where is the same as in Example 24.

Example 26. For with , , and , (51) takes the form Inequality (52) takes the form

Example 27. For , and inequality (51) takes the form
Inequality (52) takes the form
If we take Inequality (51) takes the form Inequality (52) takes the form
By replacing in (59) and in (60), we have respectively.

Example 28. For , , and The inequality (51) takes the form Inequality (52) takes the form

Remark 29. (a) In classical case for , inequalities (53) and (54) are the same as (1.7) and (1.9), also (63) and (64) are the same as (1.6) and (1.8) in [18, Corollary 1.3], respectively, while according to authors knowledge, (56), (57), (59), (60), and (61) are not existing in the literature.
(b) For , we get the reverse inequalities.

Remark 30. The results given in Section 5 can analogously be proved for .

Remark 31. The results given in Corollaries 7 and 8 and in their given applications can also be obtained for finite value of analogously.

Acknowledgments

The research of the first author is supported by a grant of the Romanian National Authority for Scientific Research, CNCS UEFISCDI, project no. PN-II-ID-PCE-2011-3-0154. The research of the second author is partially supported by the Higher Education Commission, Pakistan, and the research of the third author is supported by the Croatian Ministry of Science, Education and Sports under the research Grant 117-1170889-0888.

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