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Generalized -Convex Functions on Fractal Sets
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.
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 , 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 , 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 [4–6] 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 [7–12].
The calculus on fractal set can lead to better comprehension for the various real world models from science and engineering . Researchers have constructed many kinds of fractional calculus on fractal sets by using different approaches. Particularly, in , Yang stated the analysis of local fractional functions on fractal space systematically, which includes local fractional calculus. In , 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 , 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.
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, 16–19].
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 ). 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 ). 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 ). 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 ). Suppose that and . If , and . Suppose that , and then
Lemma 7 (see ). 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 .
Theorem 10. Let .(a)If , then is nondecreasing on and (b)If , then is nonnegative on .
Proof. (a) Since , we have, for and ,
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 ,
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 .
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 .
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,
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  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 .
From Lemmas 6 and 7, it is easy to see that is a generalized convex -type function and Moreover,
Therefore, for all .
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 .
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 ,
Let . We need only to consider .
Thus, we have
(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 .
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.
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).
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