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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 198405, 9 pages
http://dx.doi.org/10.1155/2013/198405
Research Article

Generalizations of Hölder’s and Some Related Integral Inequalities on Fractal Space

Department of Construction and Information Engineering, Guangxi Modern Vocational Technology College, Hechi, Guangxi 547000, China

Received 5 May 2013; Accepted 8 July 2013

Academic Editor: Miguel Martin

Copyright © 2013 Guang-Sheng Chen. 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

Based on the local fractional calculus, we establish some new generalizations of Hölder’s inequality. By using it, some related results on the generalized integral inequality in fractal space are investigated in detail.

1. Introduction

Let , , , and be continuous real-valued functions on . Then, the famous Hölders inequality reads as The renowned inequality of Hölder [1] is well celebrated for its beauty and its wide range of important applications to real and complex analysis and functional analysis, as well as many disciplines in applied mathematics. A large number of new proofs, various generalizations, refinements, variations, and applications of Hölder inequality have been investigated in the literature in [211]. Recently, it comes to our attention that an interesting local fractional integral Hölder’s inequality, which was established by Yang [12], is as follows.

Let , , , . Then,

Recently, the local fractional calculus has attracted a lot of interest for scientists and engineers. Local fractional derivative had been introduced in [1236]; that is the local fractional derivative was structured in [1218, 26, 3036], Jumarie gave the modified Riemann-Liouville derivative in [19, 20], the fractal derivative was considered in [2125, 2729], and the generalized fractal derivative was proposed by Chen et al. [25]. As a consequence, the theory of local fractional calculus becomes important for modelling problems for fractal mathematics and engineering on Cantor sets and it plays important role in many applications in several fields such as the theoretical physics [14, 18], the elasticity and fracture mechanics [14], the heat conduction theory [14, 16, 27], signal analysis [12, 13], the fluid mechanics [14], tensor analysis [14], Fourier and wavelet analysis [12, 13], optimization method [14], and complex analysis [12, 13]. For example, the local fractional Fokker-Planck equation was proposed in [18]. The local fractional Stieltjes transform was established in [37]. The fractal heat conduction problems were presented in [14, 27]. Local fractional improper integral was obtained in [38]. The principles of virtual work, minimum potential, and complementary energy in the mechanics of fractal media were investigated in [14]. Mean value theorems for local fractional integrals were considered in [39]. The diffusion problems in fractal media were reported in [24].

The purpose of this work is to establish some generalizations of inequality (2) and give its corresponding reverse version. Moreover, the obtained results will be applied to establish local fractional integral reverse Minkowski inequality, Dresher’s inequality, and its corresponding reverse version. This paper is divided into four sections. In Section 2, we recall some basic facts about local fractional calculus; in Section 3, we give some generalizations of the local fractional integral Hölder inequality and establish its corresponding reverse version; in Section 4, we apply the obtained results to establish reverse Minkowski inequality, Dresher’s inequality, and its reverse form involving local fractional integral; some extensions of Minkowski and Dreshers inequalities are considered also.

2. Preliminaries

In this section, we recall the basic notions of local fractional calculus (see [1214]).

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.

Lemma 1 (see [14]). Assume that is a subset of the real line and is a fractal. Let be a bi-Lipschitz mapping. Then, there exist two positive constants , and , such that for all ,

From Lemma 1, we obtain easily such that where is fractal dimension of . The result that is directly deduced from fractal geometry is related to fractal coarse-grained mass function which reads with where is dimensional Hausdorff measure.

Notice that we consider the dimensions of any fractal spaces (e.g., Cantor spaces or like-Cantor spaces) as a positive number. It looks like Euclidean space because its dimension is also a positive number. The detailed results had been considered in [1214].

Definition 2 (see [12, 14]). If there exists with , for and , then is called local fractional continuous at , denoted by . is local fractional continuous on the interval , denoted by if (9) holds for .

Definition 3 (see [13, 14]). Assume that is a nondifferentiable function of exponent , , which satisfies Hölder function of exponent , then, for such that

Definition 4 (see [13, 14]). A function is continuous of order , , or shortly continuous, if

Remark 5. Compared with (12), (9) is standard definition of local fractional continuity. Here, (11) is unified local fractional continuity [14].

2.2. Local Fractional Derivatives and Integrals

Definition 6 (see [1214]). Let . Local fractional derivative of of order at is given by where .

For any , there exists denoted by Local fractional derivative of high order is derived as and local fractional partial derivative of high order is derived as

Definition 7 (see [1214]). Let . Local fractional integral of of order in the interval is deduced by where , , and , , , is a partition of the interval .

For convenience, we assume that For any , we can get denoted by

Remark 8. If , or then we have

3. Some Generalizations of Hölder Inequality and Its Reverse Form

In the section, we give some generalizations of the inequality (2) and establish its reverse form.

Theorem 9 (reverse Hölder inequality). Let , , and let , . Then,

Proof. Set , , and then we have . By inequality (2), we obtain In (24), multiplying both sides by yields Inequality (25) implies

Combining inequality (2) and Theorem 9, we can derive the following generalization.

Corollary 10. Let , let , , and let . Then,(1)for , one has (2)for , , , one has

Proof. (1) If , , are two positive constants and . In particular, setting , , inequality (27) becomes inequality (2). Suppose (27) holds when . Using mathematical induction, let be real numbers with and , ; we must have for . In particular, we have By using Hölder’s inequality (2), we obtain since Using induction hypothesis and inequality (30), we can get Hence, this completes the proof.

(2) The Proof of (28) is similar to the proof of (27). Clearly when , inequality (28) becomes Hölder’s inequality (23). Now, suppose that (28) holds for some integer . Let , and let be such that , and let , . Note that , since

Observing Höder’s inequality (23), we have

unless .

Since Combining induction hypothesis and (34), we obtain unless for some .

4. Some Related Results

Theorem 11 (Minkowski inequality see [12]). Let , , . Then,

Next, we give reverse version of inequality (37).

Theorem 12 (reverse Minkowski's inequality). Let , , . Then,

Proof. Let By the Hölder inequality, in view of , we have By inequality (40), reverse Minkowski's inequality and the theorem are completely proved.

Corollary 13. Let , .(1)For , one has (2)For , one has

Proof. (1) Using Theorem 11, we have Multiplying to two sides of (43), we get that (41) holds.
(2) The proof of (42) is similar to the proof of (38), so we omit it here.

Corollary 14. Let , . Then,(1)for , one has (2)for , one has

Theorem 15 (Dresher’s inequality). Let , , and let ; then

Proof. Combining inequality (2) and Theorem 11, we have Using reverse Minkowski inequality (38), we have By (47) and (48), we deduce that (46) holds. This completes the proof of the theorem.

Corollary 16. Let , , . Then,

Theorem 17 (reverse Dresher’s inequality). Let , , . Then,

Proof. Let , , , , and , using Radon’s inequality (see [3]) We have if and only if sequence and sequence are proportional. Let and set . Observing (52)–(53), we have Since , let , and let , and by Theorem 12, we obtain, respectively,
Observing (54)-(55), we obtain the desired results, and the theorem is completely proved.

Corollary 18. Let , and let , . Then,

Acknowledgments

The authors would like to thank the anonymous referees for their valuable comments on the original version of this paper. This work was supported by the NNSFC (no. 11201433) and Scientific Research Project of Guangxi Education Department (no. 201204LX672).

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