A Local Fractional Integral Inequality on Fractal Space Analogous to Anderson's Inequality

Abstract

Anderson's inequality (Anderson, 1958) as well as its improved version given by Fink (2003) is known to provide interesting examples of integral inequalities. In this paper, we establish local fractional integral analogue of Anderson's inequality on fractal space under some suitable conditions. Moreover, we also show that the local fractional integral inequality on fractal space, which we have proved in this paper, is a new generalization of the classical Anderson's inequality.

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Wei Wei 1, 2 and H. M. Srivastava 3 and Yunyi Zhang 4 and Lei Wang 1 and Peiyi Shen 5 and Jing Zhang 1

Academic Editor:Xiao-Jun Yang

1, School of Computer Science and Engineering, Xi'an University of Technology, Xi'an 710048, China
2, Shaanxi Key Laboratory for Network Computing and Security Technology, Xi'an University of Technology, Xi'an 710048, China
3, Department of Mathematics and Statistics, University of Victoria, Victoria, BC, V8W 3R4, Canada
4, College of Computer and Communication Engineering, Zhengzhou University of Light Industry, Dongfeng Road, Zhengzhou, Henan Province, China
5, National School of Software, Xidian University, Xi'an 710071, China

Received 8 April 2014; Accepted 18 May 2014; 2 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

In the year 1958, Anderson [1] established the following very interesting result.

Theorem 1.

If _Fi (x) is convex increasing on [0,1] and _Fi (0)=0 for each i=1,2,...,n , then [figure omitted; refer to PDF]

Subsequently, Fink [2] improved Anderson's inequality (1) to the following form.

Theorem 2.

If _Fi (x)/x is increasing on (0,1] and _Fi (0)=0 for each i=1,2,...,n , then [figure omitted; refer to PDF]

Moreover, Fink [2] also pointed out that the condition _Fi (0)=0 (i=1,2,...,n) in Theorems 1 and 2 cannot be dropped.

In recent years, the local fractional calculus has received significantly remarkable attention from scientists and engineers. Some of the concepts of the local fractional derivative were established in [3-26]. In particular, the local fractional derivative was introduced in [3-9, 17, 21-26], Jumarie modified the Riemann-Liouville derivative in [10, 11], and the fractal derivative was proposed in [12-16, 18-20]. As a result, the theory of local fractional calculus plays an important role in applications in several different fields such as theoretical physics [5, 9], the theory of elasticity and fracture mechanics [5], and so on. For example, in [9], the authors proposed the local fractional Fokker-Planck equation. The local fractional Stieltjes transform was established in [27]. The fractal heat conduction problems were presented in [5, 18]. Local fractional improper integral was obtained in [28]. The principles of virtual work and minimum potential and complementary energy in the mechanics of fractal media were investigated in [5]. Local fractional continuous wavelet transform was studied in [29]. Mean-value theorems for local fractional integrals were considered in [30]. In [31], the authors dealt with fractal wave equations. The finite Yang-Laplace transform was introduced in [32]. Local fractional Schrödinger equation was studied in [33]. The local fractional Hilbert transform was given in [34]. The wave equation on Cantor sets was considered in [35]. The diffusion problems in fractal media were reported in [15] (see also several other recent developments on fractional calculus and local fractional calculus presented in [36-41]).

The purpose of this paper is to establish a certain local fractional integral inequality on fractal space, which is analogous to Anderson's inequality asserted by Theorem 1. This paper is divided into the following three sections. In Section 2, we recall some basic facts about local fractional calculus. In Section 3, the main result is presented.

2. Preliminaries

In this section, we would review the basic notions of local fractional calculus (see [3-5]).

2.1. Local Fractional Continuity of Functions

In order to study the local fractional continuity of nondifferentiable functions on fractal sets, we first give the following results on the local fractional continuity of functions.

Lemma 3 (see [5]).

Suppose that F is a subset of the real line and is a fractal. Suppose also that f:(F,d)[arrow right](^{Ω[variant prime]} ,^{d[variant prime]} ) is a bi-Lipschitz mapping. Then there are two positive constants ρ and τ , and F⊂R : [figure omitted; refer to PDF] such that, for all _x1 ,_x2 ∈F , [figure omitted; refer to PDF]

From Lemma 3, it is easily seen that (see [5]) [figure omitted; refer to PDF] so that [figure omitted; refer to PDF] where α is the fractal dimension of F .

Definition 4 (see [3, 5]).

Assume that there exists [figure omitted; refer to PDF] with [figure omitted; refer to PDF] for [straight epsilon],δ>0 and [straight epsilon],δ∈R . Then f(x) is said to be local fractional continuous at x=_x0 , denoted by [figure omitted; refer to PDF] The function f(x) is local fractional continuous on the interval (a,b) , denoted by (see [5]) [figure omitted; refer to PDF] if (7) holds true for x∈(a,b) .

Definition 5 (see [4, 5]).

Assume that f(x) is a nondifferentiable function of exponent α (0<α...6;1) . Then f(x) is called the Hölder function of exponent α if, for x,y∈F , one has [figure omitted; refer to PDF]

Definition 6 (see [4, 5]).

A function f(x) is said to be continuous of order α (0<α...6;1) or, equivalently, α -continuous, if [figure omitted; refer to PDF]

2.2. Local Fractional Derivatives and Local Fractional Integrals

Definition 7 (see [3-5]).

Assume that f(x)∈_Cα (a,b) . Then a local fractional derivative of f(x) of order α at x=_x0 is defined by [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

It follows from Definition 7 that there exists (see [5]) [figure omitted; refer to PDF] if [figure omitted; refer to PDF] for any x∈(a,b) .

Definition 8 (see [3, 5]).

(a) If ^f(α) (x)>0 on a given interval, then f(x) is increasing on that interval.

(b) If ^f(α) (x)<0 on a given interval, then f(x) is decreasing on that interval.

Definition 9 (see [42]).

A function f(x) is called α -convex on I if the following inequality holds true: [figure omitted; refer to PDF] for all _x1 ,_x2 ∈I and 0...6;λ...6;1 such that λ_x1 +(1-λ)_x2 ∈I .

Theorem 10 (see [42]).

Assume that f(x) is an α -local differentiable function on I . If ^f(α) (x) is nondecreasing (nonincreasing) on I , then the function f is α -convex (α -concave) on I .

Theorem 11 (see [42]).

Assume that f(x) is a local fractional continuous function. Then each of the following assertions holds true:

(1) if ^f(2α) (x) exists on I and ^f(2α) (x)...7;0 for all x∈I , then f is α -convex on I ;

(2) if ^f(2α) (x) exists on I and ^f(2α) (x)...6;0 for all x∈I , then f is α -concave on I .

Definition 12 (see [3-5]).

Assume that f(x)∈_Cα (a,b) . A local fractional integral of f(x) of order α in the interval [a,b] is expressed by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] are a partition of the interval [a,b] .

It follows from Definition 12 that (see [5]) [figure omitted; refer to PDF] if [figure omitted; refer to PDF] for any x∈(a,b) .

Remark 13 (see [3-5]).

Assume that f(x)∈^Dx(α) (a,b) or f(x)∈_Cα (a,b) ; then [figure omitted; refer to PDF]

3. Main Results

Lemma 14.

Let f(x) , g(x)∈_Cα (0,1) satisfy the constraints that f(0)=0 and g(x) is increasing on (0,1] . If the function [figure omitted; refer to PDF] is increasing on (0,1] , then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] for x∈[0,1] .

Proof.

Let [figure omitted; refer to PDF] Then, clearly, H(0)=0 and [figure omitted; refer to PDF] Moreover, we have [figure omitted; refer to PDF] Since the function [figure omitted; refer to PDF] is increasing on (0,1] , we can see that [figure omitted; refer to PDF] is decreasing on (0,1] .

Next, we show that H(x)...7;0 on [0,1] . This proof can be divided into the following two parts.

(a) Assume that there exists a point _x0 ∈(0,1) such that [figure omitted; refer to PDF]

Then [figure omitted; refer to PDF] Hence we assume that x∈[0,_x0 ] . Then H(x)...7;H(0)=0 . If we assume that x∈[_x0 ,1] , then H(x)...7;H(1)=0 on [_x0 ,1] . Thus H(x)...7;0 on [0,1] .

(b) Suppose that [figure omitted; refer to PDF]

In this case, H(x) is increasing on [0,1) . Hence H(x)...7;H(0)=0 on [0,1) . It follows from H(1)=0 that H(x)...7;0 on [0,1] . Thus, by applying a known result [5, Theorem 2.28], we have [figure omitted; refer to PDF] because H(x)...7;0 and g(x) is increasing (and hence ^g(α) (x)...7;0 ). We have thus completed our proof.

We are in a position to state and prove our main result as follows.

Theorem 15.

Let _f1 (x),_f2 (x),...,_fn (x)∈_Cα (0,1) with _fi (0)=0 and [figure omitted; refer to PDF] increasing on (0,1] for i=1,2,...,n . If _fi (x) (i=1,2,...,n ) is increasing on (0,1] , then [figure omitted; refer to PDF]

Proof.

It follows from Lemma 14 and the increasing property of _fi (x) (i=1,2,...,n) that, for ^fi* (x) defined as in Lemma 14, [figure omitted; refer to PDF] which yields [figure omitted; refer to PDF] The proof of Theorem 15 is thus completed.

Remark 16.

In its special case when α=1 , the inequality (36) asserted by Theorem 15 would reduce to the Anderson-Fink inequality (2).

Acknowledgments

This work was supported by the Scientific Research Program Funded by Shaanxi Provincial Education Department (no. 2013JK1139), the China Postdoctoral Science Foundation (no. 2013M542370), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant no. 20136118120010). This study was also supported by the National Natural Science Foundation of China (nos. 11301414, 11226173, and 61272283).

Conflict of Interests

The authors declare that they have no conflict of interests regarding the publication of this paper.

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