International Journal of AnalysisVolume 2014 (2014), Article ID 126797, 14 pageshttp://dx.doi.org/10.1155/2014/126797

Research Article

Jensen Functionals on Time Scales for Several Variables

Matloob Anwar,¹ Rabia Bibi,¹ Martin Bohner,² and Josip Pečarić³

¹School of Natural Sciences, National University of Sciences and Technology, H-12, Islamabad 44000, Pakistan²Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409-0020, USA³Faculty of Textile Technology, University of Zagreb, Pierottijeva 6, 10000 Zagreb, Croatia

Received 16 November 2013; Accepted 21 February 2014; Published 10 April 2014

Academic Editor: Baruch Cahlon

Copyright © 2014 Matloob Anwar 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 define Jensen functionals and concerned generalized means for several variables on time scales. We derive properties of Jensen functionals and apply them to generalized means. In this setting, we obtain generalizations, refinements, and conversions of many remarkable inequalities.

1. Introduction

Jensen's inequality is well known in analysis and many other areas of mathematics. Most of the classical inequalities can be obtained by using the Jensen inequality. For time scale theory, Jensen's inequality for one variable is obtained by Agarwal et al. [1], and now there are various extensions and generalizations of it given by many researchers (see [2–8]). In [3], it is shown that the Jensen inequality for one variable holds for time scale integrals including the Cauchy delta, Cauchy nabla, diamond-, Riemann, Lebesgue, multiple Riemann, and multiple Lebesgue integrals. Further, in [4], we give properties and applications of Jensen functionals on time scales for one variable.

In this paper, we obtain the Jensen inequality for several variables and deduce Jensen functionals. We discuss several properties and applications of Jensen functionals. In the sequel, we give all the results for Lebesgue delta integrals. For other time scale integrals, as mentioned above, all those results can be obtained in a similar way. These results generalize the results given in [4] for one variable. Now, we give a brief introduction of time scale integrals; for a detailed introduction we refer to [1, 9–12]. A time scale is an arbitrary closed subset of , and time scale calculus provides unification and extension of classical results. For example, when , the time scale integral is an ordinary integral, and when , the time scale integral becomes a sum. In [10, Chapter 5], the Lebesgue integral is introduced: let be a time scale interval defined by where with . Let be the Lebesgue -measure on . Suppose is a -measurable function. Then the Lebesgue -integral of on is denoted by All theorems of the general Lebesgue integration theory, including the Lebesgue dominated convergence theorem, hold also for Lebesgue -integrals on . Now, we give some properties of Lebesgue -integrals and state Jensen's inequality and Hölder's inequality for Lebesgue -integrals. Throughout this paper, denotes a time scale interval otherwise is specified.

Theorem 1 (see [3, Theorem 3.2]). If and are -integrable functions on , then

Theorem 2 (see [3, Theorem 4.2]). Assume is convex, where is an interval. Suppose is -integrable. Moreover, let be nonnegative and -integrable such that . Then

Theorem 3 (see [3, Theorem 6.2]). For , define . Let , , be nonnegative functions such that , , are -integrable on . If , then If and , or if and , then (5) is reversed.

Remark 4. Theorem 1 recalls that the Lebesgue -integral is an isotonic linear functional (see [13]). So we can also use the approach of isotonic linear functionals whenever results are known for isotonic linear functionals.

In the next section, we give Jensen inequality on time scales for several variables and define Jensen functionals. In Section 3, we investigate properties of Jensen functionals and some of its consequences regarding superadditivity and monotonicity. In Section 4, we apply these results to weighted general means, defined on time scales, and give many applications. Finally in Section 5, we give applications to Hölder's inequality on time scales.

2. Jensen Inequality and Jensen Functionals

Let be an -tuple of functions such that are -integrable on . Then denotes the -tuple: That is, -integral acts on each component of .

Theorem 5 (Jensen inequality). Assume is convex, where is closed and convex. Suppose , , are -integrable on such that for all . Moreover, let be nonnegative and -integrable such that . Then

Proof. Suppose is convex on . Therefore, for every point , there exists a point (see [13, Theorem 1.31]) such that Let . By (8), we get and hence the proof is completed.

Remark 6. By using the fact that the time scale integral is an isotonic linear functional, Theorem 5 can also be obtained by using Theorem 1 and [13, Theorem 2.6].

Definition 7. Assume , where is closed and convex. Suppose , , are -integrable on such that for all . Moreover, let be nonnegative and -integrable such that . Then one defines the Jensen functional on time scales for several variables by

Remark 8. By Theorem 5, the following statements are obvious. If is convex, then while if is concave, then

Example 9. Let , , and , in (10). Then the Jensen functional (10) becomes where with , , and . Some properties of the Jensen functional are investigated in [14, 15].

Example 10. If is a real interval, then Jensen's functional (10) becomes

3. Properties of Jensen Functionals

In the following theorem, we give our main result concerning the properties of the Jensen functional (10).

Theorem 11. Assume , where is closed and convex. Suppose , , are -integrable on such that for all . Moreover, let be nonnegative and -integrable such that and . If is convex, then is superadditive; that is, and is increasing; that is, with implies Moreover, if is concave, then is subadditive and decreasing; that is, (15) and (16) hold in reverse order.

Proof. Let be convex. Because the time scales integral is linear (see Theorem 1), it follows from Definition 7 that If , we have . Now, because Jensen's functional is superadditive and nonnegative, we have On the other hand, if is concave, then the reversed inequalities of (15) and (16) can be obtained in a similar way.

Corollary 12. Let , , , satisfy the hypotheses of Theorem 11. Further, suppose there exist nonnegative constants and such that If is convex, then while if is concave, then the inequalities in (20) hold in reverse order.

Proof. By using (10), we have Now the result follows from the second property of Theorem 11.

Corollary 13. Let , , satisfy the hypotheses of Theorem 11. Further, assume that attains its minimum value and its maximum value on its domain. If is convex, then Where Moreover, if is concave, then the inequalities in (22) hold in reverse order.

Proof. Let attain its minimum and maximum values on its domain . Then Let By using (10), we have Now the result follows from the second property of Theorem 11.

Example 14. Let the functional be defined as in Example 9. Let with and . If is convex, then Theorem 11 implies is superadditive; that is, and is increasing; that is, if such that , then Moreover, if is concave, then the inequalities in (27) and (28) hold in reverse order. If attains its minimum and maximum values on its domain, then Corollary 13 yields where if is convex. Further, the inequalities in (29) hold in reverse order if is concave.

4. Applications to Weighted Generalized Means

In the sequel, is an interval and is closed and convex.

Definition 15. Assume is strictly monotone and is a function of variables. Suppose , , are -integrable on such that for all . Moreover, let be nonnegative and -integrable such that is -integrable and . Then one defines the weighted generalized mean on time scales by

Theorem 16. Assume , , are strictly monotone and is a function of variables. Suppose , , are -integrable such that for all . Moreover, let be nonnegative and -integrable such that , , , , , are -integrable and , . If defined by is convex, then the functional is superadditive, that is, and increasing; that is, with implies Moreover, if is continuous and concave, then (33) is subadditive and decreasing; that is, (34) and (35) hold in reverse order.

Proof. The functional defined in (33) is obtained by replacing with and with , , in the Jensen functional (10) and letting ; that is, Now, all claims follow immediately from Theorem 11.

Corollary 17. Let , ,, , , , and , , satisfy the hypothesis of Theorem 16. Further, assume that attains its minimum value and its maximum value on its domain. If is convex, then where Moreover, if is concave, then the inequalities in (37) hold in reverse order.

Proof. The proof is omitted as it is similar to the proof of Corollary 13.

Remark 18. If we take the discrete form of the weighted generalized mean (31) with , then we obtain the quasiarithmetic mean. Namely, let be continuous and strictly monotone, with , , and with and . Then the quasiarithmetic mean of with weight is defined by

Now the following examples connect the quasiarithmetic mean (39) and the properties of Jensen functionals.

Example 19 (see [16, Corollary 3]). Let and be defined as in Remark 18 and let be strictly increasing and strictly convex with continuous derivatives of second order such that is concave. Further, let , , , , be defined as in Example 9, and with , , and . Then, is a convex function (see [17, Theorem 1, page 197]). Hence by Theorem 11, the functional is superadditive, that is, and increasing; that is, if such that , then Also, by Corollary 12, we have where

Example 20 (see [16, Corollary 4]). Consider (39), but with different conditions on and . Namely, if (i) for ;(ii);(iii) or ,then we define Let , , , , be defined as in Example 9 and with and . Let be strictly increasing and strictly convex with continuous derivatives of second order such that is convex. Then is a convex function (see [17, Theorem 2, page 197]). Hence, by Theorem 11, the functional is superadditive, that is, and increasing; that is, if , then Also, by Corollary 12, we have where

Example 21 (see [16, Corollary 5]). For a real-valued function defined on interval , an th order divided difference of at distinct points is defined recursively by Further, is -convex on , , if and only if, for all choices of distinct points in , Let , , , , be defined as in Example 9 and , with and . Let be -convex, where is a closed and bounded interval. Then by Theorem 11, for , the functional is superadditive, that is, and increasing; that is, if such that , then Also, by Corollary 12, we have where

Corollary 22. Assume , , and are strictly monotone. Suppose are -integrable such that for all and are nonnegative and -integrable such that , , , and , , are -integrable and , . Further, let If , , and are positive and , , and are negative, then the functional is superadditive, that is, and increasing; that is, if such that , then if and only if . If attains its minimum and maximum values on its domain , then (61) yields Moreover, if , , , , , and are all positive, then the inequalities in (60), (61), and (62) are reversed if and only if .

Proof. Let in Theorem 16. By setting , we have If , , and are positive and , , and are negative, then is convex if and only if (see [18]). If , , , , , and are all positive, then is concave if and only if (see [18]). Now, all claims follow immediately from Theorem 16.

Corollary 23. Assume , , and are strictly monotone. Suppose are -integrable such that for all and are nonnegative and -integrable such that , , , and , , are -integrable and , . Further, let If , , and are positive and , , and are negative, then the functional is superadditive, that is, and increasing; that is, if such that , then if and only if . If attains its minimum and maximum values on its domain , then (67) yields If , and are all positive, then the inequalities in (66), (67), and (68) are reversed if and only if .

Proof. Let in Theorem 16. By setting , we have If , , and are positive and , , and are negative, then is convex if and only if . If , , , , , and are all positive, then is concave if and only if (see [18]). Now, all claims follow immediately from Theorem 16.

Corollary 24. Let be such that (a), or ;(b), or , or , for ;(c), or , for .Suppose are -integrable and are nonnegative and -integrable such that , , , , , and are -integrable and , . Then the functional is superadditive, that is, and increasing; that is, if such that , then If attains its minimum and maximum values on its domain, then Moreover, the inequalities in (71), (72), and (73) are reversed provided that (a′), for ;(b′), for .

Proof. Let in Theorem 16. By setting , , , and , we have Now, is convex if and only if , which implies and these are satisfied if , , and satisfy conditions , , and . is concave if and only if , and this implies These are satisfied if , , and satisfy conditions and . Now, all claims follow immediately from Theorem 16.

Corollary 25. Let be such that , and (a), or ;(b), or , or , for ;(c), or , for .Suppose are -integrable and are nonnegative and -integrable such that , , , , , and are -integrable and , . Then the functional is superadditive, that is, and increasing; that is, if such that , then If attains its minimum and maximum values on its domain, then Moreover, the inequalities in (78), (79), and (80) are reversed provided that (a′) , for ;(b′) , for .

Proof. Let in Theorem 16. By setting , , , and , we have Now, the proof is similar to the proof of Corollary 24.

Corollary 26. Let be such that (a), for all ;(b), for ;(c), for .Suppose are -integrable and are nonnegative and -integrable such that , , , , , and are -integrable and ,. Then the functional is superadditive, that is, and increasing; that is, if such that , then If attains its minimum and maximum values on its domain, then Moreover, the inequalities in (83), (84), and (85) are reversed provided that (a′), for all ;(b′) , for ;(c′) , for .

Proof. Let in Theorem 16. By setting , , , and , we have Now, the proof is similar to the proof of Corollary 22, with some extra considerations of the definitions of , , and .

Corollary 27. Suppose are -integrable. Moreover, let be nonnegative and -integrable such that , , , and , , are -integrable and , . Then the functional is subadditive, that is, and decreasing; that is, if such that , then If attains its minimum and maximum values on its domain, then

Proof. Let in Theorem 16. By setting and , we have Now, the proof is similar to the proof of Corollary 22.

5. Applications to Hölder's Inequality

Suppose , , are nonnegative -integrable functions on such that is -integrable, where , , are such that . Then, by using Theorem 3 (Hölder's inequality on time scales), we have If , then (92) implies or In this section, we discuss properties of the functional, deduced from the Hölder inequality (93), defined in the following way.

Definition 28. Suppose is such that , , are nonnegative -integrable functions on . Let be such that and . Then one defines the functional by

Theorem 29. Let and be real -tuples with , and , . Suppose , , are nonnegative -integrable on such that and are -integrable. Then and   is increasing; that is, if such that , then

Proof. By Definition 28, we have where Now, by combining (98) and (99), we have If , then , and therefore This completes the proof.

Corollary 30. Let and satisfy the hypothesis of Theorem 29. Then

Proof. Let By Definition 28, we have Since , the result follows from the second property of Theorem 29.

Corollary 31. Let , and satisfy the hypothesis of Theorem 29 with . If there exist constants such that , then

Proof. By Definition 28, we have Now the result follows from the second property of Theorem 29.

Remark 32. Some results for isotonic linear functionals related to the results given in this paper can be found in [16].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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