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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 123913, 21 pages
http://dx.doi.org/10.1155/2012/123913
Research Article

Superadditivity, Monotonicity, and Exponential Convexity of the Petrović-Type Functionals

1Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan
2Faculty of Electrical Engineering and Computing, University of Zagreb, Zagreb 10000, Croatia
3Faculty of Textile Technology, University of Zagreb, Zagreb 10000, Croatia

Received 1 December 2011; Accepted 27 February 2012

Academic Editor: Natig Atakishiyev

Copyright © 2012 Saad Ihsan Butt 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 functionals derived from Petrović-type inequalities and establish their superadditivity, subadditivity, and monotonicity properties on the corresponding real n-tuples. By virtue of established results we also define some related functionals and investigate their properties regarding exponential convexity. Finally, the general results are then applied to some particular settings.

1. Introduction

In this paper we prove some interesting properties of the functionals derived by virtue of the Petrović and related inequalities (see, [1] pages 152–159). For the sake of simplicity these inequalities will be referred to as the Petrović-type inequalities, while the corresponding functionals will be referred to as the Petrović-type functionals.

Therefore, throughout this introduction, we present the above-mentioned Petrović-type inequalities that will be the base in our research and also define the corresponding functionals that will be the subject of our study. We start with the following inequality.

Theorem 1.1. Let be an interval, , and let be a nonnegative real -tuple such that If is such that the function is decreasing on , then In addition, if is increasing on , then the sign of inequality in (1.2) is reversed.

Remark 1.2. It should be noticed here that, if is strictly increasing function on , then the equality in (1.2) is valid if and only if we have equalities in (1.1) instead of inequalities, that is, if and .

Motivated by the above theorem, we define the Petrović-type functional , as a difference between the right-hand side and the left-hand side of inequality (1.2), that is, where , , , and is defined on the interval .

Remark 1.3. If (1.1) holds and is decreasing on , then On the other hand, if (1.1) is valid and is increasing on , then
The above functional (1.3) will also be considered under slightly altered assumptions on real -tuples and . For that sake, the following result from [1] will be used in due course.

Theorem 1.4. Suppose , is a real -tuple such that , and let . Further, let be such that is increasing on .(i)If there exists such that where , and , then (1.2) holds.(ii) If there exists such that then the reverse inequality in (1.2) holds.

Remark 1.5. If is increasing on and (1.6) holds, then the Petrović-type functional is nonnegative, that is, inequality (1.4) is valid. Conversely, if is increasing on and conditions as in (1.7) are fulfilled, then relation (1.5) holds.

In order to define another Petrović-type functional, we cite the following Petrović-type inequality involving a convex function.

Theorem 1.6. Let , and let fulfill conditions as in (1.1). If is a convex function, then

Remark 1.7. If is a concave function then is convex, hence replacing by in Theorem 1.6, we obtain inequality

Remark 1.8. If the function from Theorem 1.6 is strictly convex, then the inequality in (1.8) is strict, if all ’s are not equal or .

Now, regarding inequality (1.8) we define another Petrović-type functional by the formula provided that , , , and is defined on .

Remark 1.9. If (1.1) holds and is a convex function, then If (1.1) holds and is a concave function, then
Finally, we will also be concerned with an integral form of the Petrović-type functional, based on the following integral Petrović-type inequality.

Theorem 1.10. Let be an interval, , and let be a convex function. Further, suppose is continuous and monotone with , where is fixed, and is a function of bounded variation with (a) If and then (b) If and either there exists an such that for , or there exists an such that for , then the reverse inequality in (1.15) holds.

In view of Theorem 1.10, we define the functional which represents the integral form of the Petrović-type functional.

Remark 1.11. If the functions , , and are defined as in the statement of Theorem 1.10 and (1.14) holds, then the functional is nonnegative, that is, Moreover, if either (1.16) or (1.17) holds then
For a comprehensive inspection on the Petrović-type inequalities including proofs and diverse applications, the reader is referred to [1].
The paper is organized in the following way. After this introduction, in Section 2 we prove superadditivity, subadditivity, and monotonicity properties of functionals , and . In addition, we also derive some bounds for the functional via the nonweighted functional of the same type. By virtue of results from Section 2, in Section 3 we study some other classes of Petrović-type functionals and investigate their properties regarding exponential convexity. Finally, in Section 4 we apply our general results to some particular settings.

Convention. Throughout this paper denotes the set of real numbers, while denotes the set of nonnegative numbers (including zero). Further, bold letters , , and , respectively, denote real -tuples , , and . Moreover, means that for all .

2. Main Results

In this section we derive some interesting properties of the Petrović-type functionals , and , defined in Section 1. More precisely, we establish the conditions under which the appropriate functional is superadditive (subadditive) and increasing (decreasing), with respect to the corresponding -tuple of real numbers. Our first result refers to the Petrović-type functional defined by (1.3).

Theorem 2.1. Let , , and let nonnegative -tuples , fulfill conditions as in (1.1). If is such that the function is decreasing on , then the functional (1.3) possess the following properties.(i) is superadditive on nonnegative n-tuples, that is, provided that .(ii) If are such that and , , then that is, is increasing on nonnegative n-tuples.(iii) If is increasing on , then the signs of inequalities in (2.1) and (2.2) are reversed, that is, is subadditive and decreasing on nonnegative n-tuples.

Proof. (i) Using definition (1.3) of the Petrović-type functional and utilizing the linearity of the sum, we have On the other hand, since is decreasing function, Theorem 1.1 in the nonweighted case (for ) yields inequality Finally, combining relations (2.3) and (2.4), we obtain Therefore we have as claimed.
(ii) Monotonicity follows easily from the superadditivity property. Since , we can represent as the sum of two nonnegative -tuples, namely, . Now, from relation (2.1) we get Finally, if the conditions as in (ii) are fulfilled, then, taking into account Theorem 1.1 we have that , which implies that .
(iii) The case of increasing function is treated in the same way as in (i) and (ii), taking into account that the sign of the corresponding Petrović-type inequality is reversed.

By virtue of Theorem 1.4, the above properties of the functional can also be derived in a slightly different setting.

Theorem 2.2. Let , , and let real -tuples , fulfill conditions as in (1.6). If is such that the function is increasing on , then the functional has the following properties.(i) is superadditive on real -tuples, that is, provided that and .(ii) If , , and there exist such that where , , , and , then that is, is increasing on real -tuples.(iii) If real -tuples and fulfill conditions as in (1.7), then the signs of inequalities in (2.8) and (2.10) are reversed, that is, is subadditive and decreasing on real n-tuples.

Proof. (i) The proof follows the same lines as the proof of the previous theorem. Namely, the left-hand side of (2.8) can be rewritten as Moreover, is increasing, hence Theorem 1.4 for yields inequality Finally, relations (2.11) and (2.12) imply inequality that is, we obtain (2.8).
(ii) Considering , the real -tuple can be rewritten as . Now, regarding relation (2.8) we have Finally, taking into account conditions as in (2.9), it follows by Theorem 1.4 that , that is, , which completes the proof.
(iii) This case is treated in the same way as in (i) and (ii), taking into account that the sign of the corresponding Petrović-type inequality is reversed.

Superadditivity and monotonicity properties stated in Theorem 2.1 play an important role in numerous applications of the Petrović-type inequalities. In the sequel we utilize the monotonicity property of the Petrović-type functional . More precisely, we derive some bounds for this functional, expressed in terms of the nonweighted functional of the same type.

Corollary 2.3. Let , , and let be such that is decreasing on . Further, suppose is such that and , , where and .
If then the Petrović-type functional fulfills inequality while for one has where Moreover, if is increasing on , then the signs of inequalities in (2.15) and (2.16) are reversed.

Proof. Since , monotonicity of the Petrović-type functional implies that .
On the other hand, if is decreasing function, we have Now, regarding (2.18) we have that is, we obtain (2.15). Inequality (2.16) is derived in a similar way, by using the second inequality in (2.18).

Our next result provides superadditivity and monotonicity properties of the Petrović-type functional defined by (1.10).

Theorem 2.4. Let , , and let fulfill conditions as in (1.1). If is a convex function, then the functional (1.10) has the following properties:(i) is superadditive on nonnegative n-tuples, that is, provided that .(ii) If are such that and , then that is, is increasing on nonnegative -tuples.(iii) If is a concave function, then the signs of inequalities in (2.20) and (2.21) are reversed, that is, is subadditive and decreasing on nonnegative n-tuples.

Proof. (i) The left-hand side of inequality (2.20) can be rewritten as Further, Theorem 1.6 in the nonweighted case (for ) yields inequality hence combining relations (2.22) and (2.23), we get Thus, considering definition (1.10) we obtain (2.20), as claimed.
(ii) Monotonicity property follows from the corresponding superadditivity property (2.20), as in Theorem 2.2.
(iii) The case of concave function follows from the fact that the sign of the corresponding Petrović-type inequality is reversed.

To conclude this section we also derive the properties of the integral Petrović-type functional, defined by (1.18).

Theorem 2.5. Suppose is a convex function, is continuous and monotone with , where is fixed, and let be functions of bounded variation with Then the functional , defined by (1.18), has the following properties.(i) is subadditive with respect to functions of bounded variation, that is, where , , and .(ii) If and either there exists an such that for , for , and for , or there exists an such that for , for , and for , then

Proof. (i) Regarding definition (1.18) of the Petrović-type integral functional, we have that is, by the linearity of the differential. Now, applying inequality (1.8) to term , we obtain Further, inserting (2.30) in (2.29), we have that is, by rearranging,
(ii) Monotonicity follows from the subadditivity property (2.26). Namely, representing as , we have Clearly, under assumptions as in the statement of theorem, we have (see also Remark 1.11), hence it follows that , which completes the proof.

3. -Exponential Convexity and Exponential Convexity of the Petrović-Type Functionals

By virtue of the results from Section 2, in this section we define several new classes of Petrović-type functionals and investigate their properties regarding exponential convexity.

We start these issues by giving some definitions and notions concerning exponentially convex functions which are frequently used in the results. This is a subclass of convex functions introduced by Bernstein in [2] (see also [35]).

Definition 3.1. A function is -exponentially convex in the Jensen sense on an interval , if holds for all choices and . Function is -exponentially convex if it is -exponentially convex in the Jensen sense and continuous on .

The following remarks, propositions, and lemmas involving -exponentially convex functions are well known (see, e.g., papers [6, 7]).

Remark 3.2. It is clear from the definition that 1-exponentially convex functions in the Jensen sense are in fact nonnegative functions. Also, -exponentially convex functions in the Jensen sense are -exponentially convex in the Jensen sense for every .

By using some linear algebra and definition of the positive semidefinite matrix, we have the following proposition.

Proposition 3.3. If is an -exponentially convex in the Jensen sense then the matrix is positive semi-definite for all . In particular, for all .

Definition 3.4. A function is exponentially convex in the Jensen sense on an interval , if it is -exponentially convex in the Jensen sense for all . Moreover, function is exponentially convex if it is exponentially convex in the Jensen sense and continuous on .

Remark 3.5. It is known (and easy to show) that is a log-convex in the Jensen sense if and only if holds for each and . It follows that a function is log-convex in the Jensen sense if and only if it is 2-exponentially convex in the Jensen sense. Also, using the basic convexity theory it follows that the function is log-convex if and only if it is 2-exponentially convex.

We will also need the following result (see, e.g., [1]).

Lemma 3.6. If is a convex function on an interval and if , then the following inequality is valid: If the function is concave then the sign of the above inequality is reversed.

Divided differences are found to be very handy and interesting when we have to operate with different functions having different degree of smoothness. Let be a function, an interval in . Then for distinct points , the divided differences of first and second order are defined as follows:

The values of the divided differences are independent of the order of the points and may be extended to include the cases when some or all points are equal, that is,

provided that exists.

Now, passing through the limit and replacing by in (3.6), we have [1, page 16],

provided that exists. Also passing to the limit in (3.6), we have

provided that exists.

Remark 3.7. One can note that if for all , then is increasing on and if for all , then is convex on .

Now, we are ready to study some new classes of Petrović-type functionals. For the sake of simplicity and to avoid many notions, we first introduce the following definitions.(M1) Under the assumptions of Theorem 1.1 equipped with conditions as in (1.1), we define linear functional as(M2) Under the assumptions of Theorem 1.4 with conditions as in (1.6), we define linear functional as(M3) Under the assumptions of Theorem 1.4 with conditions as in (1.7), we define linear functional as(M4) Under the assumptions of Theorem 2.1 with conditions as in (1.1), and provided that , we define linear functional as(M5) Under the assumptions of Theorem 2.2 with conditions as in (1.6), and provided that , , we define linear functional as(M6) Under the assumptions of Theorem 2.2 with conditions as in (1.7), and provided that , , we define linear functional as(M7) Under the assumptions of Theorem 1.6 with conditions as in (1.1), we define linear functional as(M8) Under the assumptions of Theorem 1.10 with conditions as in (1.14), we define linear functional as(M9) Under the assumptions of Theorem 1.10 equipped with conditions (1.16) or (1.17), we define linear functional as(M10) Under the assumptions of Theorem 2.4 with conditions as in (1.1), and provided that , we define linear functional as(M11) Under the assumptions of Theorem 2.5, and provided that

, , , we define linear functional as

Remark 3.8. Considering the assumptions as in , , if is increasing function on then

Remark 3.9. Considering the assumptions as in , , if is convex function on then

In order to obtain our main results regarding the exponential convexity, we define different families of functions. Let be intervals. For distinct points , we define the following. and is -exponentially convex in the Jensen sense on , where . and is -exponentially convex in the Jensen sense on . and is exponentially convex in the Jensen sense on , where . and is exponentially convex in the Jensen sense on . and is 2-exponentially convex in the Jensen sense on , where . and is 2-exponentially convex in the Jensen sense on .

Theorem 3.10. Let be linear functionals defined as in , associated with families and in such a way that, , for , and , for . Then is -exponentially convex function in the Jensen sense on . If the function is continuous on , then it is -exponentially convex on .

Proof. (a) We first prove -exponential convexity in the Jensen sense of the function , for . To do this, as we have considered the families of functions defined in , for , , and , , we define the function Clearly, we have where and .
Since the function is -exponentially convex in the Jensen sense, the right-hand side of the above expression is nonnegative which implies that is increasing on (see Remark 3.7).
Hence, taking into account Remark 3.8, we have that is, Therefore, we conclude that the functions , , are -exponentially convex on in the Jensen sense.
It remains to prove the -exponential convexity in the Jensen sense of the functions , . For that sake, we consider the families of functions defined in . For each , , and , , we consider the function Obviously, Since is -exponentially convex, the right-hand side of the above expression is nonnegative which implies that is convex on . Moreover, taking into account Remark 3.9, we have that is, Hence, is -exponentially convex for , and the proof is completed.

The following corollary is an immediate consequence of the above theorem.

Corollary 3.11. Let be linear functionals defined as in , associated with families and in such a way that , , and , . Then is exponentially convex function in the Jensen sense on . If is continuous on then it is exponentially convex on .

Proof. It follows from the previous theorem.

Corollary 3.12. Let be linear functionals defined as in , associated with families and in such a way that , , and , . Then the following statements hold.(i) If the function is continuous on then it is 2-exponentially convex on and, thus, log-convex.(ii) If the function is strictly positive and differentiable on , then for all such that , one has where

Proof. (i) This is an immediate consequence of Theorem 3.10 and Remark 3.2.
(ii) By (i), the function is log-convex on , which means that the function is convex on . Hence, by using Lemma 3.6 with , , , , we obtain that is, Finally, if , by taking the limit , we have Other possible cases are treated similarly.

Remark 3.13. The results given in Theorem 3.10, Corollaries 3.11, and 3.12 are still valid when the points coincide, for a family of differentiable functions such that the function is -exponentially convex in the Jensen sense (exponentially convex in the Jensen sense, log-convex in the Jensen sense). Note also that the results given in Theorem 3.10, Corollaries 3.11, and 3.12 hold when two of the points coincide, say , for a family of differentiable functions such that the function is -exponentially convex in the Jensen sense (exponentially convex in the Jensen sense, log-convex in the Jensen sense). Moreover, the above results also hold when all three points coincide for a family of twice differentiable functions with the same property. These results can be proved easily by using the definition of divided differences and Remark 3.7.

4. Examples

We conclude this paper with several examples related to the results from the previous section.

Example 4.1. Let , and let be the function defined by Obviously, a family of functions is increasing for all , hence, by virtue of Theorem 2.1, we obtain that the functional is subadditive and decreasing on nonnegative -tuples.

Moreover, since , the mapping is exponentially convex (see [8]). Now, regarding Corollary 3.11 and Remark 3.13, we get exponential convexity of the functionals for .

In addition, Corollary 3.12 provides the log-convexity of these functionals and we have for .

Example 4.2. Suppose that and is the function defined by Since the function is increasing for every , utilizing Theorem 2.1, we obtain that the functional is subadditive and decreasing on nonnegative -tuples.
Further, since , the mapping is exponentially convex (see [8]). Now, by using Corollary 3.11 and Remark 3.13, we get exponential convexity of the functionals for .
In addition, Corollary 3.12 implies the log-convexity of these functionals and we have for .

Example 4.3. Consider the family of functions , , defined by It is easy to see that the function is increasing on for all . Hence, by virtue of Theorem 2.1, the functional is subadditive and decreasing on nonnegative -tuples.
In addition, and the mapping is exponentially convex (see [8]). Similarly as in the previous examples, Corollary 3.11 and Remark 3.13 provide exponential convexity of the functionals for .
Also, by Corollary 3.12, we get log-convexity of these functionals and we have for .

Example 4.4. Let , and let be the function defined by Obviously, a family of functions is increasing for , hence, by virtue of Theorem 2.1, we obtain that the functional is subadditive and decreasing on nonnegative -tuples.
Further, since , the mapping is exponentially convex (see [8]). Similarly as in the previous examples, regarding Corollary 3.11 and Remark 3.13, we get exponential convexity of the functionals for .
In addition, Corollary 3.12 provides the log-convexity of these functionals and we have for .

Acknowledgments

This research was partially funded by Higher Education Commission, Pakistan. The research of the second and third authors was supported by the Croatian Ministry of Science, Education and Sports under the Research Grants 036-1170889-1054 and 117-1170889-0888.

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