Journal of Function Spaces and Applications

Volume 2012 (2012), Article ID 384045, 17 pages

doi:10.1155/2012/384045

Research Article

Jensen's Functionals on Time Scales

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

¹Centre for Advanced Mathematics and Physics, National University of Sciences and Technology, Islamabad 44000, Pakistan

²Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409-0020, USA

³Faculty of Textile Technology, University of Zagreb, Pierottijeva 6, 10000 Zagreb, Croatia

Received 5 July 2012; Accepted 6 September 2012

Academic Editor: Ti-Jun Xiao

Copyright © 2012 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 consider Jensen’s functionals on time scales and discuss its properties and applications. Further, we define weighted generalized and power means on time scales. By applying the properties of Jensen’s functionals on these means, we obtain several refinements and converses of Hölder’s inequality on time scales.

1. Introduction

Time scales theory was initiated by Hilger [1], and now there is a lot of work in this field. For an introduction to the theory of dynamic equations on time scales, we refer to [2–4]. Time scales calculus provides unification, extension, and generalization of classical continuous and discrete results. In this paper, we give results only for Lebesgue -integrals, but all the results obtained are also true if we take instead certain other time scales integrals such as the Cauchy delta, Cauchy nabla, -diamond, multiple Riemann, or multiple Lebesgue integral.

Now, using the same notations as in [4, Chapter 5], we briefly give an introduction of Lebesgue -integrals. Let be a time scales interval defined by

Suppose is the Lebesgue -measure on and 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 . The following theorem compares the Lebesgue -integral with the Riemann -integral.

Theorem 1.1 (see [4, Theorem  5.81]). Let be a closed bounded interval in , and let be a bounded real-valued function defined on . If is Riemann -integrable from to , then is Lebesgue -integrable on , and

where and indicate the Riemann and Lebesgue integrals, respectively.

The results in this paper are based on the authors’ results given in [5]. For related results we refer the reader to [6, 7]. The remaining theorems in this section are taken from [5]. Theorem 1.2 shows that the Lebesgue -integral is a so-called isotonic linear functional. Theorem 1.3 recalls Jensen’s inequality for Lebesgue -integrals, while Theorem 1.4 states Hölder’s inequality for Lebesgue -integrals. These three results are used in the remainder of this paper.

Theorem 1.2 (see [5, Theorem 3.2]). If and are -integrable functions on , then

Theorem 1.3 (see [5, Theorem 4.2]). Assume is convex, where is an interval. Suppose is -integrable. Moreover, let be nonnegative and -integrable such that . Then

Theorem 1.4 (see [5, Theorem 6.2]). For , define . Let , , be nonnegative functions such that , , are -integrable on . If , then

If and , or if and , then (1.6) is reversed.

In Section 2, we define Jensen’s functionals and, by using Jensen’s inequality on time scales (Theorem 1.3), give some of their properties concerning superadditivity and monotonicity. In Section 3, we apply the properties of Jensen’s functionals to generalized means, defined on time scales, and obtain improvements of several classical inequalities on time scales. Finally, in Section 4, we give applications of Hölder’s inequality on time scales (Theorem 1.4) and obtain several refinements and converses of this inequality.

2. Properties of Jensen’s Functionals

Definition 2.1 (Jensen’s functional). Assume , where is an interval. Suppose is -integrable. Moreover, let be nonnegative and -integrable such that . Then we define Jensen’s functional on time scales by

Remark 2.2. By Theorem 1.3, the following statements are obvious. If is convex, then

while if is concave, then

Theorem 2.3. Assume , where is an interval. Suppose is -integrable. Also, 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, (2.4) and (2.5) hold in reverse order.

Proof. Let be convex. Because the time scales integral is linear (see Theorem 1.2), it follows from Definition 2.1 that

If , we have . Now, because Jensen’s functional is superadditive (see above) and nonnegative (see Theorem 1.2), we have

On the other hand, if is concave, then the reversed inequalities of (2.4) and (2.5) can be obtained in a similar way.

Corollary 2.4. Let , , , satisfy the hypotheses of Theorem 2.3. Further, suppose there exist nonnegative constants and such that

If is convex, then

while if is concave, then the inequalities in (2.9) hold in reverse order.

Proof. By using Definition 2.1, we have

Now the result follows from the second property of Theorem 2.3.

Corollary 2.5. Let , , satisfy the hypotheses of Theorem 2.3. 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 (2.11) hold in reverse order.

Proof. Let attain its minimum value and its maximum value on its domain . Then

By Definition 2.1, we have

Now the result follows from the second property of Theorem 2.3.

Remark 2.6. The first inequality in (2.11) gives a converse of Jensen’s inequality on time scales, and the second inequality in (2.11) gives a refinement of the observed inequality.

Example 2.7 (see [8, Remark 4]). Let us take the discrete form of Jensen’s functional (2.1). For this, let , , , , and , for . Then (2.1) becomes

where

Under these notations, (2.11) takes the form

where

Example 2.8 (see [8, Remark 5]). In addition to the notation introduced in Example 2.7, let for and put . Using

in Corollary 2.4, (2.9) becomes

Example 2.9. Suppose and . Then Jensen’s functional (2.1) becomes

3. Applications to Weighted Generalized Means

Definition 3.1 (weighted generalized mean). Assume is strictly monotone, where is an interval. Suppose is -integrable. Also, let be nonnegative and -integrable such that . Then we define the weighted generalized mean on time scales by

provided (3.1) is well defined.

Theorem 3.2. Assume are strictly monotone, where is an interval. Suppose is -integrable. Moreover, let be nonnegative and -integrable such that the functional

is well defined. If is convex, then (3.2) is superadditive, that is,

and (3.2) is increasing, that is, with implies

Moreover, if is concave, then (3.2) is subadditive and decreasing, that is, (3.3) and (3.4) hold in reverse order.

Proof. The functional defined in (3.2) is obtained by replacing with and with in Jensen’s functional (2.1), that is,

Now, all claims follow immediately from Theorem 2.3.

Corollary 3.3. Let , , , satisfy the hypotheses of Theorem 3.2. 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 (3.6) hold in reverse order.

Proof. The proof is omitted as it is similar to the proof of Corollary 2.5.

Definition 3.4 (weighted generalized power mean). Assume . Suppose is positive and -integrable, where is an interval. Moreover, let be nonnegative and -integrable such that . Then we define the weighted generalized power mean on time scales by

provided (3.8) is well defined.

Remark 3.5. The weighted generalized power mean defined in (3.8) follows from the weighted generalized mean defined in (3.1) by taking () in the weighted generalized mean.

Theorem 3.6. Assume with . Suppose is positive and -integrable, where is an interval. Moreover, let be nonnegative and -integrable such that the functional

is well defined. If , then (3.9) is superadditive (also if ), that is,

and (3.9) is increasing, that is, with implies

Moreover, if or , then (3.9) is subadditive and decreasing, that is, (3.10) and (3.11) hold in reverse order.

Proof. If , then let and () in Theorem 3.2. Then and therefore

Thus is convex if and concave if or . If, however, , then let and () in Theorem 3.2. Then . Thus is convex for . In either case the result follows now immediately from Theorem 3.2.

Corollary 3.7. Let , , , satisfy the hypotheses of Theorem 3.6. Further, assume that attains its minimum value and its maximum value on its domain. If , then

where

Moreover, if or , then the inequalities in (3.13) hold in reverse order.

Proof. The proof is omitted as it is similar to the proof of Corollary 2.5 followed by Theorem 3.6.

Example 3.8 (see [8, Remark 7]). From the discrete form of Corollary 3.7, that is, by using , we get a refinement and a converse of the arithmetic-geometric mean inequality. Using the notation as introduced in Example 2.7, let for all and , . Then (3.13) becomes

where

The first inequality in (3.15) gives a converse and the second one gives a refinement of the arithmetic geometric-mean inequality of and .

Theorem 3.9. Let , , , satisfy the hypotheses of Theorem 3.6. Suppose that the functional

is well defined. If , then (3.17) is superadditive, that is,

and (3.17) is increasing, that is, with implies

Moreover, if , then (3.17) is subadditive and decreasing, that is, (3.18) and (3.19) hold in reverse order.

Proof. Let and in Theorem 3.2. Then . Thus is convex if and concave if . Now the rest of the proof follows immediately from Theorem 3.2.

Corollary 3.10. Let , , satisfy the hypotheses of Theorem 3.6. Further, assume that attains its minimum value and its maximum value on its domain. If , then

where is defined in (3.14). Moreover, if , then the inequalities in (3.20) hold in reverse order.

Proof. The proof is omitted as it is similar to the proof of Corollary 2.5 followed by Theorem 3.9.

Example 3.11 (see [8, Remark 8]). Again we consider . Using the notation as introduced in Example 3.8, the term takes the form

and (3.20) becomes

The inequalities in (3.22) provide a refinement and a converse of the arithmetic-geometric mean inequality in quotient form.

Example 3.12 (see [8, Remark 9]). The relations (3.15) and (3.22) also yield refinements and converses of Young’s inequality. To see this, consider again . Using the notation as introduced in Example 3.8, define

where and are positive -tuples such that . Then, (3.15) and (3.22) become

The inequalities in (3.24) and (3.25) provide the refinements and converses of Young’s inequality in difference and quotient form.

4. Improvements of Hölder’s Inequality

Let and let be -integrable for all . Assume for all are conjugate exponents, that is, , and is -integrable on . Hölder’s inequality on time scales (Theorem 1.4) asserts that

Theorem 4.1. Let , , be conjugate exponents, and let , , be nonnegative -integrable functions such that and are nonnegative and -integrable. Then the following inequalities hold:

Proof. Let , , in Example 3.12. Then the expressions in (3.24) become

Now, by applying the -integral to the last two equations, we get

By applying the -integral to the series of inequalities in (3.24), we obtain the required inequalities.

Remark 4.2. The first inequality in Theorem 4.1 gives a converse and the second one gives a refinement of Hölder’s inequality on time scales.

Theorem 4.3. Under the same assumption as in Theorem 4.1, the following inequalities hold:

provided that all expressions are well defined.

Proof. We consider relation (3.25) in the same settings as in Theorem 4.1. By inverting, (3.25) can be rewritten in the form

Now, if we consider the -tuple , where

then the expressions that represent the means in (4.6) become

Now, by taking the -integral on (4.6) in described setting, we obtain the required inequalities.

Remark 4.4. The first inequality in Theorem 4.3 gives a refinement and the second one gives a converse of Hölder’s inequality on time scales.

Corollary 4.5. Let such that . Further, assume that , are positive and -integrable such that attains its minimum value and its maximum value on its domain. If , then

Moreover, if , then the inequalities in (4.9) hold in reverse order.

Proof. The result follows from Corollary 2.5 by replacing with , with , and letting . Then is convex on , and we have

If , then by substituting and in (2.11), we get (4.9). If , then , and since the expressions and contain the factor , we conclude that the inequalities in (4.9) hold in reverse order in that case.

Remark 4.6. The first inequality in (4.9) gives a converse and the second one gives a refinement of Hölder’s inequality on time scales.

Corollary 4.7. Let such that and . Further, assume that , are positive and -integrable such that attains its minimum value and its maximum value on its domain. Then

Proof. In Corollary 2.5, replace with , with , and let . Then is convex on . We get

Now, the result follows immediately from (2.11).

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