#### 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
*