Generalized Convex Functions on Fractal Sets and Two Related Inequalities

Abstract

We introduce the generalized convex function on fractal sets [superscript]Rα[/superscript] (0<α...4;1) 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.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Academic Editor:Praveen Agarwal

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

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

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.

1. Introduction

Let f:I⊆R[arrow right]R . For any _x1 ,_x2 ∈I and λ∈[0,1] , if the following inequality [figure omitted; refer to PDF] holds, then f is called a convex function on I .

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 f is a convex function on [a,b] . Then, for any _xi ∈[a,b] and _λi ∈[0,1] (i=1,2,...,n) with ^∑i=1n_λi =1 , we have [figure omitted; refer to PDF]

Hermite-Hadamard's Inequality [4]. Let f be a convex function on [a,b] with a<b . If f is integral on [a,b] , then [figure omitted; refer to PDF]

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 [5-9]. Many researchers studied the properties of functions on fractal space and constructed many kinds of fractional calculus by using different approaches (see [10-14]). 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 f is concave if and only if -f 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 s ets. In Section 5, some applications are given on fractal sets by means of the generalized Jensen's inequality.

2. Preliminaries

Recall the set ^Rα 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 0<α...4;1 , we have the following α -type set of element sets.

: ^Zα : the α -type set of the integer is defined as the set {^0α ,±^1α ,±^2α ,...,±^nα ,...} .

: ^Qα : the α -type set of the rational numbers is defined as the set {^mα =^(p/q)α :p∈Z,q...0;0} .

: ^Jα : the α -type set of the irrational numbers is defined as the set {^mα ...0;^(p/q)α :p∈Z,q...0;0} .

: ^Rα : the α -type set of the real line numbers is defined as the set ^Rα =^Qα ∪^Jα .

If ^aα ,^bα , and ^cα belong to the set ^Rα of real line numbers, then one has the following:

(1) ^aα +^bα and ^aαbα belong to the set ^Rα ;

(2) ^aα +^bα =^bα +^aα =(a+b^)α =(b+a^)α ;

(3) ^aα +(^bα +^cα )=(a+b^)α +^cα ;

(4) ^aαbα =^bαaα =(ab^)α =(ba^)α ;

(5) ^aα (^bαcα )=(^aαbα )^cα ;

(6) ^aα (^bα +^cα )=^aαbα +^aαcα ;

(7) ^aα +^0α =^0α +^aα =^aα and ^aα1α =^1αaα =^aα .

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

Definition 1 (see [13]).

A nondifferentiable function f:R[arrow right]^Rα , x[arrow right]f(x) is called local fractional continuous at _x0 , if, for any [straight epsilon]>0 , there exists δ>0 , such that [figure omitted; refer to PDF] holds for |x-_x0 |<δ , where [straight epsilon],δ∈R . If f(x) is local fractional continuous on the interval (a,b) , one denotes f(x)∈_Cα (a,b) .

Definition 2 (see [13]).

The local fractional derivative of f(x) of order α at x=_x0 is defined by [figure omitted; refer to PDF] where ^Δα (f(x)-f(_x0 ))=Γ(1+α)(f(x)-f(_x0 )) .

If there exists ^f((k+1)α) (x)= ^Dxα ...^Dxα ...k+1 timesf(x) for any x∈I⊆R , then one denotes f∈_D(k+1)α (I) , where k=0,1,2,... .

Definition 3 (see [13]).

The local fractional integral of the function f(x) of order α is defined by [figure omitted; refer to PDF] with Δ_tj =_tj+1 -_tj and Δt=max...{Δ_tj |"j=1,2,...,N-1} , where [_tj ,_tj+1 ] , j=0,...,N-1 , and _t0 =a<_t1 <...<_ti <...<_tN-1 <_tN =b is a partition of the interval [a,b] .

Here, it follows that ^Iaa(α) f(x)=0 if a=b and ^Iab(α) f(x)=-^Iba(α) f(x) if a<b . If, for any x∈[a,b] , there exists ^Iax(α) f(x) , then it is denoted by f(x)∈^Ix(α) [a,b] .

Lemma 4 (see [13] generalized local fractional Taylor theorem).

Suppose that ^f(k+1)α (x)∈_Cα (I) , for interval I⊆R , k=0,1,...,n , 0<α...4;1 . And let _x0 ∈[a,b] . Then, for any x∈I , there exists at least one point ξ , which lies between the points x and _x0 , such that [figure omitted; refer to PDF]

Remark 5.

When I⊆R is an open interval (a,b) , 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 I⊆R , the formula is also true.

3. Generalized Convex Functions

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

Definition 6.

Let f:I⊆R[arrow right]^Rα . For any _x1 , _x2 ∈I and λ∈[0,1] , if the following inequality [figure omitted; refer to PDF] holds, then f is called a generalized convex function on I .

Definition 7.

Let f:I[arrow right]^Rα . For any _x1 ...0;_x2 ∈I and λ∈[0,1] , if the following inequality [figure omitted; refer to PDF] holds, then f is called a generalized strictly convex function on I⊆R .

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 f is called a generalized concave function or generalized strictly concave function, respectively.

Here are two basic examples of generalized strictly convex functions:

(1) f(x)=^xαp , x...5;0 , p>1 ;

(2) f(x)=_Eα (^xα ) , x∈R , where _Eα (^xα )=^∑k=0∞ (^xαk /Γ(1+kα)) is the Mittag-Leffler function.

Note that the linear function f(x)=^aαxα +^bα , x∈R is generalized convex and also generalized concave.

We will focus our attention on the convexity since a function f is concave if and only if -f 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 f:I[arrow right]^Rα . Then f is a generalized convex function if and only if the inequality [figure omitted; refer to PDF] holds, for any _x1 ,_x2 ,_x3 ∈I with _x1 <_x2 <_x3 .

Proof.

In fact, taking λ=(_x3 -_x2 )/(_x3 -_x1 ) , then _x2 =λ_x1 +(1-λ)_x3 . And by the generalized convexity of f , we get [figure omitted; refer to PDF]

From the above formula, it is easy to see that [figure omitted; refer to PDF]

Reversely, for any two points _x1 ,_x3 (_x1 <_x3 ) on I⊆R , we take _x2 =λ_x1 +(1-λ)_x3 for λ∈(0,1) . Then _x1 <_x2 <_x3 and λ=(_x3 -_x2 )/(_x3 -_x1 ) . Using the above inverse process, we have [figure omitted; refer to PDF]

So, f is a convex function on I⊆R .

In the same way, it can be shown that f is a generalized convex function on I⊆R if and only if [figure omitted; refer to PDF] for any _x1 ,_x2 ,_x3 ∈I with _x1 <_x2 <_x3 .

Theorem 9.

Letting f∈_Dα (I) , then the following conditions are equivalent:

(1) f is a generalized convex function on I ,

(2) ^f(α) is an increasing function on I ,

(3) for any _x1 ,_x2 ∈I , [figure omitted; refer to PDF]

Proof.

( 1 [arrow right] 2 ) Let _x1 ,_x2 ∈I with _x1 <_x2 . And take h>0 which is small enough such that _x1 -h,_x2 +h∈I . Since _x1 -h<_x1 <_x2 <_x2 +h , then using Theorem 8 we have [figure omitted; refer to PDF]

Since f∈_Dα (I) , then letting h[arrow right]⁰⁺ , it follows that [figure omitted; refer to PDF]

So, ^f(α) is increasing in I .

( 2 [arrow right] 3 ) Take _x1 ,_x2 ∈I . Without loss of generality, we can assume that _x1 <_x2 . Since ^f(α) is increasing in the interval I , then applying the generalized local fractional Taylor theorem, we have [figure omitted; refer to PDF] where ξ∈(_x1 ,_x2 ) . That is to say, [figure omitted; refer to PDF]

( 3 [arrow right] 1 ) For any _x1 , _x2 ∈I , we let _x3 =λ_x1 +(1-λ)_x2 , where 0<λ<1 . It is easy to see that _x1 -_x3 =(1-λ)(_x1 -_x2 ) and _x2 -_x3 =λ(_x2 -_x1 ) . Then from the third condition, we have [figure omitted; refer to PDF] At the above two formulas, multiply ^λα and (1-λ^)α , respectively; then we obtain [figure omitted; refer to PDF]

So f is a generalized convex function on I .

Corollary 10.

Let f∈_D2α (a,b) . Then f is a generalized convex function (or a generalized concave function) if and only if [figure omitted; refer to PDF] for any x∈(a,b) .

4. Some Inequalities

Theorem 11 (generalized Jensen's inequality).

Assume that f is a generalized convex function on [a,b] . Then for any _xi ∈[a,b] and _λi ∈[0,1] (i=1,2,...,n) with ^∑i=1n_λi =1 , we have [figure omitted; refer to PDF]

Proof.

When n=2 , the inequality is obviously true. Assume that for n=k the inequality is also true. Then, for any _x1 ,_x2 ,...,_xk ∈[a,b] and _γi >0 , i=1,2,...,k , with ^∑i=1k_γi =1 , we have [figure omitted; refer to PDF]

If _x1 ,_x2 ,...,_xk ,_x(k+1) ∈[a,b] and _λi >0 for i=1,2,...,k+1 with ^∑i=1k+1_λi =1 , then one sets up _γi =_λi /(1-_λk+1 ) , i=1,2,...,k . It is easy to see ^∑i=1k_γi =1 .

Thus, [figure omitted; refer to PDF]

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

Corollary 12.

Let f∈_D2α [a,b] and ^f(2α) (x)...5;0 for any x∈[a,b] . Then for any _xi ∈[a,b] and _λi ∈[0,1] (i=1,2,...,n) with ^∑i=1n_λi =1 we have [figure omitted; refer to PDF]

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 ^aα/pbα/q ...4;(^aα /p)(^bα /q) , where ^aα ,^bα >0 , p,q...5;1 , and (1/p)+(1/q)=1 .

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 |_ak |>0 , |_bk |>0 , k=1,2,...,n . Then we have [figure omitted; refer to PDF]

Proof.

Take f(x)=^x2α . It is easy to see that ^f(2α) (x)...5;0 for any x∈(a,b) .

Take [figure omitted; refer to PDF] Then 0...4;_λk ...4;1 (k=1,2,...,n) with ^∑k=1n_λk =1 .

Thus, by Jensen's inequality f(^∑k=1n_λkxk )...4;^{∑k=1nλk(α)} f(_xk ) , we have [figure omitted; refer to PDF] The above formula can be reduced to [figure omitted; refer to PDF] which implies that [figure omitted; refer to PDF]

Thus we have [figure omitted; refer to PDF]

Theorem 14 (generalized Hermite-Hadamard's inequality).

Let f(x)∈^Ix(α) [a,b] be a generalized convex function on [a,b] with a<b . Then [figure omitted; refer to PDF]

Proof.

Let x=a+b-y . Then [figure omitted; refer to PDF]

Furthermore, when x∈[(a+b)/2,b] , a+b-x∈[a,(a+b)/2] . And by the convexity of f , we have [figure omitted; refer to PDF]

Thus [figure omitted; refer to PDF]

For another part, we first note that if f is a generalized convex function, then, for t∈[0,1] , it yields [figure omitted; refer to PDF]

By adding these inequalities we have [figure omitted; refer to PDF] Then, integrating the resulting inequality with respect to t over [0,1] , we obtain [figure omitted; refer to PDF]

It is easy to see that [figure omitted; refer to PDF] So, [figure omitted; refer to PDF]

Combining the inequalities (36) and (41), we have [figure omitted; refer to PDF]

Note that it will be reduced to the class Hermite-Hadamard inequality if α=1 .

5. Applications of Generalized Jensen's Inequality

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

Example 15.

Let a>0 , b>0 and ^a3α +^b3α ...4;^2α . Then a+b...4;2 .

Proof.

Let f(x)=^x3α , x∈(0,+∞) . It is easy to see that f is a generalized convex function.

So, [figure omitted; refer to PDF] That is, [figure omitted; refer to PDF]

Thus, we conclude that a+b...4;2 .

Example 16.

Let x,y∈R . Then [figure omitted; refer to PDF] where _Eα (^xα )=^∑k=0∞ (^xαk /Γ(1+kα)) is the Mittag-Leffler function.

Proof.

Take f(x)=_Eα (^xα ) . It is easy to see ^{(_Eα (xα ))(2α)} =_Eα (^xα )>0 . So, the generalized Jensen's inequality gives [figure omitted; refer to PDF]

Example 17 (power mean inequality).

Let _a1 ,_a2 ,...,_an >0 and 0<s<t or s<t<0 . Denote [figure omitted; refer to PDF]

Then _Ss ...4;_St . And _Ss =_St if and only if _a1 =_a2 =...=_an .

Proof.

Consider the following.

Case I (0<s<t ). Take f(x)=^x(t/s)α , x>0 . Then [figure omitted; refer to PDF] By the generalized Jensen's inequality, we have [figure omitted; refer to PDF] That is, [figure omitted; refer to PDF] From the above formula, it is easy to see [figure omitted; refer to PDF]

So, we have _Ss ...4;_St .

Case II ( s < t < 0 ) . Letting _bi =1/^ai and applying the case for 0<-t<-s , we can get the conclusion.

Example 18.

If a,b,c>0 and a+b+c=1 , then find the minimum of [figure omitted; refer to PDF] Solution. Note that 0<a,b,c<1 . Let f(x)=^(x+1/x)10α , x∈(0,1) . Then, via the formula [figure omitted; refer to PDF] we have [figure omitted; refer to PDF]

By the generalized Jensen's inequality, [figure omitted; refer to PDF]

So, the minimum is 1^010α /^39α , when a=b=c=1/3.

Example 19.

If a,b,c,d>0 and ^c2α +^d2α =(^a2α +^b2α)3 , then [figure omitted; refer to PDF]

Proof.

Let _x1 =^{(a3 /c)1/2} , _x2 =^{(b3 /d)1/2} , _y1 =(ac^)1/2 , _y2 =(bd^)1/2 . By the generalized Cauchy-Schwartz inequality, we have [figure omitted; refer to PDF]

Canceling ^aαcα +^bαdα 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.

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).

Conflict of Interests

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

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