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Abstract and Applied Analysis

Volume 2014 (2014), Article ID 254737, 8 pages

http://dx.doi.org/10.1155/2014/254737
Research Article

Generalized -Convex Functions on Fractal Sets

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

Received 5 May 2014; Revised 27 June 2014; Accepted 27 June 2014; Published 17 July 2014

Academic Editor: Hari M. Srivastava

Copyright © 2014 Huixia Mo and Xin Sui. 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 two kinds of generalized -convex functions on real linear fractal sets . And similar to the class situation, we also study the properties of these two kinds of generalized -convex functions and discuss the relationship between them. 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, such as in biological system, economy, and optimization [1, 2]. In [3], Hudzik and Maligranda generalized the definition of convex function and considered, among others, two kinds of functions which are -convex.

Let and , and then the two kinds of -convex functions are defined, respectively, in the following way.

Definition 1. A function, , is said to be -convex in the first sense if for all and all with . One denotes this by .

Definition 2. A function, , is said to be -convex in the second sense if for all and all with . One denotes this by .

It is obvious that the -convexity means just the convexity when , no matter whether it is in the first sense or in the second sense. In [3], some properties of -convex functions in both senses are considered and various examples and counterexamples are given. There are many research results related to the -convex functions; see [46] and so on.

In recent years, the fractal 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 [712].

The calculus on fractal set can lead to better comprehension for the various real world models from science and engineering [8]. Researchers have constructed many kinds of fractional calculus on fractal sets by using different approaches. Particularly, in [13], Yang stated the analysis of local fractional functions on fractal space systematically, which includes local fractional calculus. In [14], the authors introduced the generalized convex function on fractal sets and established the generalized Jensen inequality and generalized Hermite-Hadamard inequality related to generalized convex function. And, in [15], Wei et al. established a local fractional integral inequality on fractal space analogous to Anderson’s inequality for generalized convex functions. The generalized convex function on fractal sets can be stated as follows.

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

Inspired by these investigations, we will introduce the generalized -convex function in the first or second sense on fractal sets and study the properties of generalized -convex functions.

The paper is organized as follows. In Section 2, we state the operations with real line number fractal sets and give the definitions of the local fractional calculus. In Section 3, we introduce the definitions of two kinds of generalized -convex functions and study the properties of these functions. In Section 4, we give some applications for the two kinds of generalized -convex functions on fractal sets.

2. Preliminaries

Let us recall the operations with real line number on fractal space and use Gao-Yang-Kang’s idea to describe the definitions of the local fractional derivative and local fractional integral [13, 1619].

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 3 (see [13]). A nondifferentiable function , is called to be 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 4 (see [13]). The local fractional derivative of function of order at is defined by where and the Gamma function is defined by .

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

Definition 5 (see [13]). Let . Then the local fractional integral of the function of order is defined by with , , and , , where is a partition of the interval .

Lemma 6 (see [13]). Suppose that and . If , and . Suppose that , and then

Lemma 7 (see [13]). Suppose that ; then

3. Generalized -Convexity Functions

The convexity of functions plays a significant role in many fields. In this section, let us introduce two kinds of generalized -convex functions on fractal sets. And then, we study the properties of the two kinds of generalized -convex functions.

Definition 8. Let . A function is said to be generalized -convex in the first sense, if for all and all with . One denotes this by .

Definition 9. A function is said to be generalized -convex in the second sense, if for all and all with . One denotes this by .

Note that, when , the generalized -convex functions in both senses are the generalized convex functions; see [14].

Theorem 10. Let .(a)If , then is nondecreasing on and (b)If , then is nonnegative on .

Proof. (a) Since , we have, for and , The function is continuous on , decreasing on , and increasing on and . This yields that for and . If , then . Therefore, by the fact that (15) holds, we get for all . So we can obtain that

So, taking , we get which means that is nondecreasing on .

As for the second part, for and with , we have And taking , we get So,

(b) For , we can get that, for ,

So, . This means that , since .

Remark 11. The above results do not hold, in general, in the case of generalized convex functions, that is, when , because a generalized convex function, , need not be either nondecreasing or nonnegative.

Remark 12. If , then the function is nondecreasing on but not necessarily on .

Function is called to be generalized convex in each variable, if For all ,   and with .

Theorem 13. Let . If and and if is a generalized convex and nondecreasing function in each variable, then the function defined by is in . In particular, if , then , .

Proof. If , then for all with , Thus, .

Moreover, since ,   are nondecreasing generalized convex functions on , so they yield particular cases of our theorem.

Let us pay attention to the situation when the condition    in the definition of can be equivalently replaced by the condition    .

Theorem 14. (a) Let . Then inequality (10) holds for all and all with if and only if .

(b) Let . Then inequality (11) holds for all and all with if and only if .

Proof. (a) Necessity is obvious by taking and . Let us show the sufficiency.

Assume that and with . Put and . Then and

(b) Necessity. Taking , we obtain . And using Theorem 10(b), we get . Therefore .

Sufficiency. Let and with . Put and , and then .

So,

Theorem 15. (a) Let . If and , then .

(b) Let . If and , then .

(c) Let . If and , then .

Proof. (a) Assume that and . Let with , and we have . From Theorem 14(b), we can get for , and then .

(b) Assume that , , and with . Then we have So .

(c) Assume that , , and with . Then . Thus, according to Theorem 14(a), we have So, .

Theorem 16. Let and be a nondecreasing function. Then the function defined for by belongs to .

Proof. Let and with . We consider two cases.

Case I. It is easy to see that is a nondecreasing function. Let , and then

Case II. Let , and then . So, and . Thus, that is,

Thus, we can get that Then,

We obtain

Theorem 17. (a) Let and , where , . If is a nondecreasing function and is a nonnegative function such that and , then the composition of with belongs to , where .

(b) Let and , where , . Assume that , . If and are nonnegative functions such that either and , or and , then the product of and belongs to , where .

Proof. (a) Let ,  with , where . Since for , then according to Theorem 3(a) in [3] and Theorem 14(a) in the paper, we have which means that .

(b) According to Theorem 10(a), are nondecreasing on .

So, or, equivalently, for all .

If , then the inequality is still true because are nonnegative and either and or and .

Now let and with , where . Then for . And by Theorem 14(a), we have which means that .

Remark 18. From the above proof, we can get that if is a nondecreasing function in and is a nonnegative convex function on , then the composition of with belongs to .

Remark 19. Generalized convex functions on need not be monotonic. However, if and are nonnegative, generalized convex and either both are nondecreasing or both are nonincreasing on , then the product is also a generalized convex function.

Let be a continuous function. Then is said to be a -function if and is nondecreasing on . Similarly, we can define the -type function on fractal sets as follows. A function is said to be a -type function if and is nondecreasing.

Corollary 20. If is a generalized convex -type function and is a -function, then the composition belongs to . In particular, the -type function belongs to .

Corollary 21. If is a convex -function and is a -type function, then the composition belongs to . In particular, the -type function belongs to .

Theorem 22. If and is a -type function, then there exists a generalized convex -type function such that for all .

Proof. By the generalized -convexity of the function and by , we obtain for all and all .

Assume now that . Then that is,

Inequality (44) means that the function is a nondecreasing function on . And, since is a -type function, thus is local fractional continuous .

Define

From Lemmas 6 and 7, it is easy to see that is a generalized convex -type function and Moreover,

Therefore, for all .

4. Applications

Based on the properties of the two kinds of generalized -convex functions in the above section, some applications are given.

Example 23. Let , and . For , define

We have the following conclusions.(i)If and , then .(ii)If and , then is nondecreasing on but not on .(iii)If and , then .(iv)If and , then .

Proof. (i) Let with . Then, there are two nontrivial cases.

Case I. Let . Then .

Thus,

Case II. Let . We need only to consider .

Thus, we have So, .

(ii) From Theorem 10, we can see that property (ii) is true.

(iii) Let with . Similar to the estimate of (i), there are also two cases.

Let . Then ,

Thus,

Let . We need only to consider .

Thus, we have

So, .

(iv) Assume that , and then is nonnegative on . On the other hand, we can take , such that , which contradict the assumption.

Example 24. Let and . For , define The function is nonnegative, not local fractional continuous at and belongs to but not to .

Proof. From Theorem 16, we have that . In the following, let us show that is not in .

Take an arbitrary and put . Consider all such that , where and .

If , it must be for all and all .

Define the function Then the function is local fractional continuous on the and

It is easy to see that is local fractional continuous on and . So there is a number , , such that . The local fractional continuity of yields that for a certain , that is, inequality (55) does not hold, which means that .

5. Conclusion

In the paper, we introduce the definitions of two kinds of generalized -convex function on fractal sets and study the properties of these generalized -convex functions. When , these results are the classical situation.

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. And, this work is supported by the National Natural Science Foundation of China (no. 11161042).

References

  1. J. R. Jonathan and P. A. Mattew, “Jensen's inequality predicts effects of environmental variation,” Trends in Ecology & Evolution, vol. 14, no. 9, pp. 361–366, 1999.
  2. M. Grinalatt and J. T. Linnainmaa, “Jensen's inequality, parameter uncertainty, and multiperiod investment,” Review of Asset Pricing Studies, vol. 1, no. 1, pp. 1–34, 2011. View at Publisher · View at Google Scholar
  3. H. Hudzik and L. Maligranda, “Some remarks on s-convex functions,” Aequationes Mathematicae, vol. 48, no. 1, pp. 100–111, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. E. Set, “New inequalities of Ostrowski type for mappings whose derivatives are s-convex in the second sense via fractional integrals,” Computers & Mathematics with Applications, vol. 63, no. 7, pp. 1147–1154, 2012.
  5. M. Avci, H. Kavurmaci, and M. Emin Özdemir, “New inequalities of Hermite-HADamard type via s-convex functions in the second sense with applications,” Applied Mathematics and Computation, vol. 217, no. 12, pp. 5171–5176, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  6. Y. Shuang, H. Yin, and F. Qi, “Hermite-Hadamard type integral inequalities for geometric-arithmetically s-convex functions,” Analysis, vol. 33, no. 2, pp. 197–208, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  7. K. M. Kolwankar and A. D. Gangal, “Local fractional calculus: a calculus for fractal space-time,” in Fractals: Theory and Applications in Engineering, pp. 171–181, Springer, London, Uk, 1999. View at MathSciNet
  8. A. K. Golmankhaneh and D. Baleanu, “On a new measure on fractals,” Journal of Inequalitiesand Applications, vol. 522, no. 1, pp. 1–9, 2013.
  9. B. B. Mandelbrot, The Fractal Geometry of Nature, Macmillan, New York, NY, USA, 1983.
  10. K. J. Falconer, Fractal Geometry: Mathematical Foundations and Applications, John Wiley & Sons, New York, NY, USA, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  11. G. A. Edgar, Integral Probability and Fractal Measures, Springer, Berlin, Germany, 1998.
  12. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematical Studies, Elsevier Science, North-Holland, Amsterdam, The Netherlands, 2006.
  13. X. J. Yang, Advanced Local Fractional Calculus and Its Applications, World Science, New York, NY, USA, 2012.
  14. H. Mo, X. Sui, and D. Yu, “Generalized convex functions on fractal sets and two related inequalities,” Abstract and Applied Analysis, vol. 2014, Article ID 636751, 7 pages, 2014. View at Publisher · View at Google Scholar
  15. W. Wie, H. M. Srivastava, Y. Zhang, L. Wang, P. Shen, and J. Zhang, “A local fractional integral inequality on fractal space analogous to Anderson's inequality,” Abstract and Applied Analysis, vol. 2014, Article ID 797561, 7 pages, 2014. View at Publisher · View at Google Scholar
  16. X. J. Yang, D. Baleanu, and J. A. T. Machado, “Mathematical aspects of heisenberg uncertainty principle within local fractional fourier analysis,” Boundary Value Problems, vol. 2013, no. 1, pp. 131–146, 2013.
  17. Y. Zhao, D. F. Cheng, and X. J. Yang, “Approximation solutions for local fractional Schrödinger equation in the on-dimensional cantorian system,” Advances in Mathematical Physics, vol. 2013, Article ID 291386, 5 pages, 2013. View at Publisher · View at Google Scholar
  18. A.-M. Yang, Z.-S. Chen, H. M. Srivastava, and X.-J. Yang, “Application of the local fractional series expansion method and the variational iteration method to the Helmholtz equation involving local fractional derivative operators,” Abstract and Applied Analysis, vol. 2013, Article ID 259125, 6 pages, 2013. View at Publisher · View at Google Scholar
  19. X. J. Yang, D. Baleanu, Y. Khan, and S. T. Mohyud -Din, “Local fractional variational iteration method for diffusion and wave equations on Cantor sets,” Romanian Journal of Physics, vol. 59, no. 1-2, pp. 36–48, 2014. View at MathSciNet