Abstract and Applied Analysis

Volume 2014 (2014), Article ID 797561, 7 pages

http://dx.doi.org/10.1155/2014/797561

Research Article

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

Wei Wei,1,2 H. M. Srivastava,³ Yunyi Zhang,⁴ Lei Wang,¹ Peiyi Shen,⁵ and Jing Zhang¹

¹School of Computer Science and Engineering, Xi’an University of Technology, Xi’an 710048, China

²Shaanxi Key Laboratory for Network Computing and Security Technology, Xi’an University of Technology, Xi’an 710048, China

³Department of Mathematics and Statistics, University of Victoria, Victoria, BC, Canada, V8W 3R4

⁴College of Computer and Communication Engineering, Zhengzhou University of Light Industry, Dongfeng Road, Zhengzhou, Henan Province, China

⁵National School of Software, Xidian University, Xi’an 710071, China

Received 8 April 2014; Accepted 18 May 2014; Published 2 June 2014

Academic Editor: Xiao-Jun Yang

Copyright © 2014 Wei Wei 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.

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.

1. Introduction

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

Theorem 1. If is convex increasing on and for each , then

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

Theorem 2. If is increasing on and for each , then

Moreover, Fink [2] also pointed out that the condition    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 is a subset of the real line and is a fractal. Suppose also that is a bi-Lipschitz mapping. Then there are two positive constants and , and :

such that, for all ,

From Lemma 3, it is easily seen that (see [5])

so that

where is the fractal dimension of .

Definition 4 (see [3, 5]). Assume that there exists

with

for and . Then is said to be local fractional continuous at , denoted by

The function is local fractional continuous on the interval , denoted by (see [5])

if (7) holds true for .

Definition 5 (see [4, 5]). Assume that is a nondifferentiable function of exponent . Then is called the Hölder function of exponent if, for , one has

Definition 6 (see [4, 5]). A function is said to be continuous of order or, equivalently, -continuous, if

2.2. Local Fractional Derivatives and Local Fractional Integrals

Definition 7 (see [3–5]). Assume that . Then a local fractional derivative of of order at is defined by

where

It follows from Definition 7 that there exists (see [5])

if

for any .

Definition 8 (see [3, 5]). (a) If on a given interval, then is increasing on that interval.

(b) If on a given interval, then is decreasing on that interval.

Definition 9 (see [42]). A function is called -convex on if the following inequality holds true:

for all and such that .

Theorem 10 (see [42]). Assume that is an -local differentiable function on . If is nondecreasing (nonincreasing) on , then the function is -convex (-concave) on .

Theorem 11 (see [42]). Assume that is a local fractional continuous function. Then each of the following assertions holds true

(1)if exists on and for all , then is -convex on ;(2)if exists on and for all , then is -concave on .

Definition 12 (see [3–5]). Assume that . A local fractional integral of of order in the interval is expressed by

where

are a partition of the interval .

It follows from Definition 12 that (see [5])

if

for any .

Remark 13 (see [3–5]). Assume that or ; then

3. Main Results

Lemma 14. Let , satisfy the constraints that and is increasing on . If the function

is increasing on , then

where

for .

Proof. Let

Then, clearly, and

Moreover, we have

Since the function

is increasing on , we can see that

is decreasing on .

Next, we show that on . This proof can be divided into the following two parts.

(a)Assume that there exists a point such that

Then

Hence we assume that . Then . If we assume that , then on . Thus on .

(b)Suppose that

In this case, is increasing on . Hence on . It follows from that on . Thus, by applying a known result [5, Theorem 2.28], we have

because and is increasing (and hence ). We have thus completed our proof.

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

Theorem 15. Let with and

increasing on for . If   () is increasing on , then

Proof. It follows from Lemma 14 and the increasing property of    that, for defined as in Lemma 14,

which yields

The proof of Theorem 15 is thus completed.

Remark 16. In its special case when , the inequality (36) asserted by Theorem 15 would reduce to the Anderson-Fink inequality (2).

Conflict of Interests

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

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).

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