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Abstract and Applied Analysis

Volume 2014, Article ID 636751, 7 pages

http://dx.doi.org/10.1155/2014/636751
Research Article

Generalized Convex Functions on Fractal Sets and Two Related Inequalities

School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China

Received 25 March 2014; Revised 20 May 2014; Accepted 30 May 2014; Published 15 June 2014

Academic Editor: Praveen Agarwal

Copyright © 2014 Huixia Mo 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 introduce the generalized convex function on fractal sets of real line numbers and study the properties of the generalized convex function. Based on these properties, we establish the generalized Jensen’s inequality and generalized Hermite-Hadamard's inequality. Furthermore, some applications are given.

1. Introduction

Let . For any and , if the following inequality holds, then is called a convex function on .

The convexity of functions plays a significant role in many fields, for example, in biological system, economy, optimization, and so on [1, 2]. And many important inequalities are established for the class of convex functions. For example, Jensen’s inequality and Hermite-Hadamard’s inequality are the best known results in the literature, which can be stated as follows.

Jensen’s Inequality [3]. Assume that is a convex function on . Then, for any and with , we have

Hermite-Hadamard’s Inequality [4]. Let be a convex function on with . If is integral on , then

In recent years, the fractal theory has received significantly remarkable attention from scientists and engineers. In the sense of Mandelbrot, a fractal set is the one whose Hausdorff dimension strictly exceeds the topological dimension [59]. Many researchers studied the properties of functions on fractal space and constructed many kinds of fractional calculus by using different approaches (see [1014]). Particularly, in [13], Yang stated the analysis of local fractional functions on fractal space systematically, which includes local fractional calculus and the monotonicity of function.

Inspired by these investigations, we will introduce the generalized convex function on fractal sets and establish the generalized Jensen’s inequality and generalized Hermite-Hadamard’s inequality related to generalized convex function. We will focus our attention on the convexity since a function is concave if and only if is convex. So, every result for the convex function can be easily restated in terms of concave functions.

The paper is organized as follows. In Section 2, we state the operations with real line number on fractal sets and give the definitions of the local fractional derivatives and local fractional integral. In Section 3, we introduce the definition of the generalized convex function on fractal sets and study the properties of the generalized convex functions. In Section 4, we establish the generalized Jensen’s inequality and generalized Hermite-Hadamard’s inequality on fractal sets. In Section 5, some applications are given on fractal sets by means of the generalized Jensen’s inequality.

2. Preliminaries

Recall the set of real line numbers and use Gao-Yang-Kang’s idea to describe the definitions of the local fractional derivative and local fractional integral.

Recently, the theory of Yang’s fractional sets [13] was introduced as follows.

For , we have the following -type set of element sets. : the -type set of the integer is defined as the set . : the -type set of the rational numbers is defined as the set . : the -type set of the irrational numbers is defined as the set . : the -type set of the real line numbers is defined as the set .

If , and belong to the set of real line numbers, then one has the following:(1) and belong to the set ;(2) ; (3) ; (4) ; (5) ; (6) ; (7) and .

Let us now state some definitions about the local fractional calculus on .

Definition 1 (see [13]). A nondifferentiable function , is called local fractional continuous at , if, for any , there exists , such that holds for , where . If is local fractional continuous on the interval , one denotes .

Definition 2 (see [13]). The local fractional derivative of of order at is defined by where .

If there exists for any , then one denotes , where .

Definition 3 (see [13]). The local fractional integral of the function of order is defined by with and , where , , and is a partition of the interval .

Here, it follows that if and if . If, for any , there exists , then it is denoted by .

Lemma 4 (see [13] generalized local fractional Taylor theorem). Suppose that , for interval , , . And let . Then, for any , there exists at least one point , which lies between the points and , such that

Remark 5. When is an open interval , Yang [13] has given the proof for the generalized local fractional Taylor theorem. In fact, using the generalized local fractional Lagrange’s theorem and following the proof of the class Taylor theorem, we can show that, for any interval , the formula is also true.

3. Generalized Convex Functions

From an analytical point of view, we have the following definition.

Definition 6. Let . For any , and , if the following inequality holds, then is called a generalized convex function on .

Definition 7. Let . For any and , if the following inequality holds, then is called a generalized strictly convex function on .

It follows immediately, from the given definitions, that a generalized strictly convex function is also generalized convex. But, the converse is not true. And if these two inequalities are reversed, then is called a generalized concave function or generalized strictly concave function, respectively.

Here are two basic examples of generalized strictly convex functions:(1) , , ;(2) , , where is the Mittag-Leffler function.

Note that the linear function , is generalized convex and also generalized concave.

We will focus our attention on the convexity since a function is concave if and only if is convex. So, every result for the convex function can be easily restated in terms of concave functions.

In the following, we will study the properties of the generalized convex functions.

Theorem 8. Let . Then is a generalized convex function if and only if the inequality holds, for any with .

Proof. In fact, taking , then . And by the generalized convexity of , we get

From the above formula, it is easy to see that

Reversely, for any two points on , we take for . Then and . Using the above inverse process, we have

So, is a convex function on .

In the same way, it can be shown that is a generalized convex function on if and only if for any with .

Theorem 9. Letting , then the following conditions are equivalent:(1) is a generalized convex function on ,(2) is an increasing function on ,(3)for any ,

Proof. Let with . And take which is small enough such that . Since , then using Theorem 8 we have

Since , then letting , it follows that

So, is increasing in .

Take . Without loss of generality, we can assume that . Since is increasing in the interval , then applying the generalized local fractional Taylor theorem, we have where . That is to say,

For any , , we let , where . It is easy to see that and . Then from the third condition, we have At the above two formulas, multiply and , respectively; then we obtain

So is a generalized convex function on .

Corollary 10. Let . Then is a generalized convex function (or a generalized concave function) if and only if for any .

4. Some Inequalities

Theorem 11 (generalized Jensen’s inequality). Assume that is a generalized convex function on . Then for any and with , we have

Proof. When , the inequality is obviously true. Assume that for the inequality is also true. Then, for any and , , with , we have

If and for with , then one sets up , . It is easy to see .

Thus,

So, the mathematical induction gives the proof of Theorem 11.

Corollary 12. Let and for any . Then for any and with we have

Using the generalized Jensen’s inequality and the convexity of functions, we can also get some integral inequalities.

In [13], Yang established the generalized Cauchy-Schwarz’s inequality by the estimate , where , , and .

Now, via the generalized Jensen’s inequality, we will give another proof for the generalized Cauchy-Schwarz’s inequality.

Corollary 13 (generalized Cauchy-Schwarz’s inequality). Let , , . Then we have

Proof. Take . It is easy to see that for any .

Take Then with .

Thus, by Jensen’s inequality , we have The above formula can be reduced to which implies that

Thus we have

Theorem 14 (generalized Hermite-Hadamard’s inequality). Let be a generalized convex function on with . Then

Proof. Let . Then

Furthermore, when , . And by the convexity of , we have

Thus

For another part, we first note that if is a generalized convex function, then, for , it yields

By adding these inequalities we have Then, integrating the resulting inequality with respect to over , we obtain

It is easy to see that So,

Combining the inequalities (36) and (41), we have

Note that it will be reduced to the class Hermite-Hadamard inequality if .

5. Applications of Generalized Jensen’s Inequality

Using the generalized Jensen’s inequality, we can get some inequalities.

Example 15. Let , and . Then .

Proof. Let , . It is easy to see that is a generalized convex function.

So, That is,

Thus, we conclude that .

Example 16. Let . Then where is the Mittag-Leffler function.

Proof. Take . It is easy to see . So, the generalized Jensen’s inequality gives

Example 17 (power mean inequality). Let and or . Denote

Then . And if and only if .

Proof. Consider the following.

Case I ( ). Take , . Then By the generalized Jensen’s inequality, we have That is, From the above formula, it is easy to see

So, we have

Case II . Letting and applying the case for , we can get the conclusion.

Example 18. If and , then find the minimum of Solution. Note that . Let , . Then, via the formula we have

By the generalized Jensen’s inequality,

So, the minimum is , when

Example 19. If and , then

Proof. Let , , , . By the generalized Cauchy-Schwartz inequality, we have

Canceling on both sides, we get the desired result.

6. Conclusion

In the paper, we introduce the definition of generalized convex function on fractal sets. Based on the definition, we study the properties of the generalized convex functions and establish two important inequalities: the generalized Jensen’s inequality and generalized Hermite-Hadamard’s inequality. At last, we also give some applications for these inequalities on fractal sets.

Conflict of Interests

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

Acknowledgments

The authors would like to express their gratitude to the reviewers for their very valuable comments. This work is supported by the National Natural Science Foundation of China (no. 11161042).

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