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.

1. Introduction

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

Theorem 1. If 𝐹_𝑖(𝑥) is convex increasing on [0,1] and 𝐹_𝑖(0)=0 for each 𝑖=1,2,…,𝑛, then ¹₀𝐹₁(𝑥)𝐹₂(𝑥)⋯𝐹_𝑛(𝑥)≧d𝑥2𝑛𝑛+1¹₀𝐹₁(𝑥)d𝑥⋯¹₀𝐹_𝑛(𝑥)d𝑥. (1)

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

Theorem 2. If 𝐹_𝑖(𝑥)/𝑥 is increasing on (0,1] and 𝐹_𝑖(0)=0 for each 𝑖=1,2,…,𝑛, then ¹₀𝐹₁(𝑥)𝐹₂(𝑥)⋯𝐹_𝑛(𝑥)≧2d𝑥_𝑛𝑛+1¹₀𝐹₁(𝑥)d𝑥⋯¹₀𝐹_𝑛(𝑥)d𝑥. (2)

Moreover, Fink [2] also pointed out that the condition 𝐹_𝑖(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)→(Ω^′,𝑑^′) is a bi-Lipschitz mapping. Then there are two positive constants 𝜌 and 𝜏, and 𝔽⊂ℝ: 𝜌^𝑠𝐻^𝑠(𝔽)≦𝐻^𝑠𝑓(𝔽)≦𝜏^𝑠𝐻^𝑠(𝔽)(𝔽⊂ℝ), (3) such that, for all 𝑥₁,𝑥₂∈𝔽, 𝜌^𝛼𝑥₁−𝑥₂^𝛼≦𝑓𝑥₁−𝑓𝑥₂≦𝜏^𝛼𝑥₁−𝑥₂^𝛼. (4)

From Lemma 3, it is easily seen that (see [5]) 𝑓𝑥₁−𝑓𝑥₂≦𝜏^𝛼𝑥₁−𝑥₂^𝛼𝑥₁,𝑥₂∈𝔽, (5) so that 𝑓𝑥₁−𝑓𝑥₂≦𝜀^𝛼𝑥₁,𝑥₂∈𝔽, (6) where 𝛼 is the fractal dimension of 𝔽.

Definition 4 (see [3, 5]). Assume that there exists 𝑓(𝑥)−𝑓𝑥₀≦𝜀^𝛼, (7) with 𝑥−𝑥₀^𝛼≦𝛿^𝛼 (8) for 𝜀,𝛿>0 and 𝜀,𝛿∈ℝ. Then 𝑓(𝑥) is said to be local fractional continuous at 𝑥=𝑥₀, denoted by lim_𝑥→𝑥0𝑓(𝑥)=𝑓𝑥₀. (9) The function 𝑓(𝑥) is local fractional continuous on the interval (𝑎,𝑏), denoted by (see [5]) 𝑓(𝑥)∈𝐶_𝛼(𝑎,𝑏) (10) 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≦1). Then 𝑓(𝑥) is called the Hölder function of exponent 𝛼 if, for 𝑥,𝑦∈𝔽, one has 𝑓(𝑥)−𝑓𝑦≦𝐶𝑥−𝑦^𝛼(0<𝛼≦1). (11)

Definition 6 (see [4, 5]). A function 𝑓(𝑥) is said to be continuous of order 𝛼(0<𝛼≦1) or, equivalently, 𝛼-continuous, if 𝑓(𝑥)−𝑓𝑥₀≦𝑜𝑥−𝑥₀^𝛼(0<𝛼≦1). (12)

2.2. Local Fractional Derivatives and Local Fractional Integrals

Definition 7 (see [3–5]). Assume that 𝑓(𝑥)∈𝐶_𝛼(𝑎,𝑏). Then a local fractional derivative of 𝑓(𝑥) of order 𝛼 at 𝑥=𝑥₀ is defined by 𝑓^(𝛼)𝑥₀=𝑑^𝛼𝑓(𝑥)𝑑𝑥^𝛼_𝑥=𝑥0=lim_𝑥→𝑥0Δ^𝛼𝑓(𝑥)−𝑓𝑥₀𝑥−𝑥₀^𝛼, (13) where Δ^𝛼𝑓(𝑥)−𝑓𝑥₀≅Γ(1+𝛼)Δ𝑓(𝑥)−𝑓𝑥₀. (14)
It follows from Definition 7 that there exists (see [5]) 𝑓(𝑥)∈𝐷_𝑥^(𝛼)(𝑎,𝑏) (15) if 𝑓^(𝛼)(𝑥)=𝐷_𝑥^(𝛼)𝑓(𝑥) (16) 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: 𝑓𝜆𝑥₁+(1−𝜆)𝑥₂≦𝜆^𝛼𝑓𝑥₁+(1−𝜆)^𝛼𝑓𝑥₂ (17) for all 𝑥₁,𝑥₂∈𝐼 and 0≦𝜆≦1 such that 𝜆𝑥₁+(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)≧0 for all 𝑥∈𝐼, then 𝑓 is 𝛼-convex on 𝐼;(2)if 𝑓^(2𝛼)(𝑥) exists on 𝐼 and 𝑓^(2𝛼)(𝑥)≦0 for all 𝑥∈𝐼, then 𝑓 is 𝛼-concave on 𝐼.

Definition 12 (see [3–5]). Assume that 𝑓(𝑥)∈𝐶_𝛼(𝑎,𝑏). A local fractional integral of 𝑓(𝑥) of order 𝛼 in the interval [𝑎,𝑏] is expressed by _𝑎𝐼_𝑏^(𝛼)𝑓(𝑥)=1Γ(1+𝛼)^𝑏_𝑎𝑓(𝑡)(d𝑡)^𝛼=1Γ(1+𝛼)lim_Δ𝑡→0^𝑁−1_𝑗=0𝑓𝑡_𝑗Δ𝑡_𝑗^𝛼, (18) where Δ𝑡_𝑗=𝑡_𝑗+1−𝑡_𝑗𝑗=0,1,𝑁−1,Δ𝑡=maxΔ𝑡₁,Δ𝑡₂,…,Δ𝑡_𝑗,𝑡,…_𝑗,𝑡_𝑗+1𝑗=0,1,…,𝑁−1𝑡₀=𝑎;𝑡_𝑁=𝑏 (19) are a partition of the interval [𝑎,𝑏].
It follows from Definition 12 that (see [5]) 𝑓(𝑥)∈_𝑎𝐼_𝑥^(𝛼)(𝑎,𝑏), (20) if _𝑎𝐼_𝑥^(𝛼)𝑓(𝑥) (21) for any 𝑥∈(𝑎,𝑏).

Remark 13 (see [3–5]). Assume that 𝑓(𝑥)∈𝐷_𝑥^(𝛼)(𝑎,𝑏) or 𝑓(𝑥)∈𝐶_𝛼(𝑎,𝑏); then 𝑓(𝑥)∈_𝑎𝐼_𝑥^(𝛼)(𝑎,𝑏). (22)

3. Main Results

Lemma 14. Let 𝑓(𝑥), 𝑔(𝑥)∈𝐶_𝛼(0,1) satisfy the constraints that 𝑓(0)=0 and 𝑔(𝑥) is increasing on (0,1]. If the function 𝑓(𝑥)Γ(1+2𝛼)𝑥^𝛼/Γ(1+𝛼) (23) is increasing on (0,1], then 1Γ(1+𝛼)¹₀𝑓(𝑥)𝑔(𝑥)(d𝑥)^𝛼≧1Γ(1+𝛼)¹₀𝑓^∗(𝑥)𝑔(𝑥)(d𝑥)^𝛼, (24) where 𝑓^∗(𝑥)=Γ(1+2𝛼)𝑥^𝛼Γ()⋅11+𝛼Γ(1+𝛼)¹₀𝑓(𝑢)(d𝑢)^𝛼 (25) for 𝑥∈[0,1].

Proof. Let 𝐻(𝑥)=1Γ(1+𝛼)^𝑥₀𝑓^∗(𝑡)−𝑓(𝑡)(d𝑡)^𝛼(𝑥∈[0,1]). (26) Then, clearly, 𝐻(0)=0 and 𝐻(1)=1Γ(1+𝛼)¹₀𝑓^∗(𝑡)−𝑓(𝑡)(d𝑡)^𝛼=1Γ(1+𝛼)¹₀𝑓^∗(𝑡)(d𝑡)^𝛼−1Γ(1+𝛼)¹₀𝑓(𝑡)(d𝑡)^𝛼=1Γ(1+𝛼)¹₀Γ(1+2𝛼)𝑡^𝛼Γ()⋅11+𝛼Γ(1+𝛼)¹₀𝑓(𝑢)(d𝑢)^𝛼(d𝑡)^𝛼−1Γ(1+𝛼)¹₀𝑓(𝑢)(d𝑢)^𝛼=1Γ(1+𝛼)¹₀Γ(1+2𝛼)𝑡^𝛼Γ()()1+𝛼d𝑡^𝛼×1Γ(1+𝛼)¹₀𝑓(𝑢)(d𝑢)^𝛼−1Γ(1+𝛼)¹₀𝑓(𝑢)(d𝑢)^𝛼=Γ(1+2𝛼)Γ()⋅Γ()𝑡1+𝛼1+𝛼^2𝛼Γ(1+2𝛼)¹₀−1^𝛼×1Γ(1+𝛼)¹₀𝑓(𝑢)(d𝑢)^𝛼=𝑡^2𝛼1₀−1^𝛼1Γ(1+𝛼)¹₀𝑓(𝑢)(d𝑢)^𝛼=0. (27) Moreover, we have 𝐻^(𝛼)(0)=𝑓^∗(0)−𝑓(0)=𝑓^∗(0)𝐻(𝑥)=0,=𝑓^∗(𝑥)(𝑥)=Γ()𝑥−𝑓1+2𝛼^𝛼Γ()⋅11+𝛼Γ(1+𝛼)¹₀𝑓(𝑢)(d𝑢)^𝛼(𝑥)=Γ()𝑥−𝑓1+2𝛼^𝛼Γ()×11+𝛼Γ(1+𝛼)¹₀𝑓(𝑢)(d𝑢)^𝛼−𝑓(𝑥)Γ(1+2𝛼)𝑥^𝛼()((./Γ1+𝛼𝑥∈0,1]) (28) Since the function 𝑓(𝑥)Γ(1+2𝛼)𝑥^𝛼/Γ(1+𝛼) (29) is increasing on (0,1], we can see that 𝐻^(𝛼)(𝑥)Γ(1+2𝛼)𝑥^𝛼/Γ(1+𝛼) (30) is decreasing on (0,1].
Next, we show that 𝐻(𝑥)≧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 𝐻^(𝛼)𝑥₀Γ(1+2𝛼)𝑥^𝛼₀/Γ(1+𝛼)=0. (31)Then 𝐻^(𝛼)(𝑥)≧0𝑥∈0,𝑥₀,𝐻^(𝛼)(𝑥)≦0𝑥∈𝑥₀,1. (32) Hence we assume that 𝑥∈[0,𝑥₀]. Then 𝐻(𝑥)≧𝐻(0)=0. If we assume that 𝑥∈[𝑥₀,1], then 𝐻(𝑥)≧𝐻(1)=0 on [𝑥₀,1]. Thus 𝐻(𝑥)≧0 on [0,1].(b)Suppose that 𝐻^(𝛼)(𝑥)Γ(1+2𝛼)𝑥^𝛼()((./Γ1+𝛼>0𝑥∈0,1)) (33)In this case, 𝐻(𝑥) is increasing on [0,1). Hence 𝐻(𝑥)≧𝐻(0)=0 on [0,1). It follows from 𝐻(1)=0 that 𝐻(𝑥)≧0 on [0,1]. Thus, by applying a known result [5, Theorem 2.28], we have 1Γ(1+𝛼)¹₀𝑓(𝑥)−𝑓^∗(𝑥)𝑔(𝑥)(d𝑥)^𝛼1=−Γ(1+𝛼)¹₀𝐻^(𝛼)(𝑥)𝑔(𝑥)(d𝑥)^𝛼=−𝐻^(𝛼)(𝑥)𝑔(𝑥)¹₀−1Γ(1+𝛼)¹₀𝑔^(𝛼)(𝑥)𝐻(𝑥)(d𝑥)^𝛼=1Γ(1+𝛼)¹₀𝑔^(𝛼)(𝑥)𝐻(𝑥)(d𝑥)^𝛼≧0, (34) because 𝐻(𝑥)≧0 and 𝑔(𝑥) is increasing (and hence 𝑔^(𝛼)(𝑥)≧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),…,𝑓_𝑛(𝑥)∈𝐶_𝛼(0,1) with 𝑓_𝑖(0)=0 and 𝑓_𝑖(𝑥)Γ(1+2𝛼)𝑥^𝛼/Γ(1+𝛼) (35) increasing on (0,1] for 𝑖=1,2,…,𝑛. If 𝑓_𝑖(𝑥)  (𝑖=1,2,…,𝑛) is increasing on (0,1], then 1Γ(1+𝛼)¹₀^𝑛_𝑖=1𝑓_𝑖(𝑥)(d𝑥)^𝛼≧[Γ(1+2𝛼)]^𝑛[Γ(1+𝛼)]^𝑛1Γ(1+𝛼)¹₀𝑥^𝑛𝛼(d𝑥)^𝛼×^𝑛_𝑖=11Γ(1+𝛼)¹₀𝑓_𝑖(𝑢)(d𝑢)^𝛼. (36)

Proof. It follows from Lemma 14 and the increasing property of 𝑓_𝑖(𝑥)  (𝑖=1,2,…,𝑛) that, for 𝑓^∗_𝑖(𝑥) defined as in Lemma 14, 1Γ(1+𝛼)¹₀^𝑛_𝑖=1𝑓_𝑖(𝑥)(d𝑥)^𝛼=1Γ(1+𝛼)¹₀𝑓_𝑛(𝑥)^𝑛−1_𝑖=1𝑓_𝑖(𝑥)(d𝑥)^𝛼≧1Γ(1+𝛼)¹₀𝑓^∗_𝑛(𝑥)^𝑛−1_𝑖=1𝑓_𝑖(𝑥)(d𝑥)^𝛼=1Γ()×1+𝛼₁₀Γ(1+2𝛼)𝑥^𝛼Γ()⋅11+𝛼Γ(1+𝛼)¹₀𝑓_𝑛(𝑢)(d𝑢)^𝛼×^𝑛−1_𝑖=1𝑓_𝑖(𝑥)(d𝑥)^𝛼=1Γ(1+𝛼)¹₀Γ(1+2𝛼)𝑥^𝛼Γ(1+𝛼)^𝑛−1_𝑖=1𝑓_𝑖(𝑥)(d𝑥)^𝛼=1Γ(1+𝛼)¹₀𝑓_𝑛(𝑢)(d𝑢)^𝛼×1Γ(1+𝛼)¹₀𝑓_𝑛−1(𝑥)Γ(1+2𝛼)𝑥^𝛼Γ(1+𝛼)^𝑛−2_𝑖=1𝑓_𝑖(𝑥)(d𝑥)^𝛼≧1Γ(1+𝛼)¹₀𝑓_𝑛(𝑢)(d𝑢)^𝛼×1Γ(1+𝛼)¹₀𝑓^∗_𝑛−1(𝑥)Γ(1+2𝛼)𝑥^𝛼Γ(1+𝛼)^𝑛−2_𝑖=1𝑓_𝑖(𝑥)(d𝑥)^𝛼=1Γ(1+𝛼)¹₀𝑓_𝑛(𝑢)(d𝑢)^𝛼×1Γ(1+𝛼)¹₀𝑓_𝑛−1(𝑢)(d𝑢)^𝛼⋅1Γ(1+𝛼)¹₀[Γ(1+2𝛼)]²𝑥^2𝛼[Γ(1+𝛼)]^2𝑛−2_𝑖=1𝑓_𝑖(𝑥)(d𝑥)^𝛼=1Γ(1+𝛼)¹₀𝑓_𝑛(𝑢)(d𝑢)^𝛼×1Γ(1+𝛼)¹₀𝑓_𝑛−1(𝑢)(d𝑢)^𝛼⋅1Γ(1+𝛼)¹₀[Γ(1+2𝛼)]²𝑥^2𝛼[Γ(1+𝛼)]²𝑓_𝑛−2(𝑥)^𝑛−3_𝑖=1𝑓_𝑖(𝑥)(d𝑥)^𝛼≧1Γ(1+𝛼)¹₀𝑓_𝑛(𝑢)(d𝑢)^𝛼×1Γ(1+𝛼)¹₀𝑓_𝑛−1(𝑢)(d𝑢)^𝛼×1Γ(1+𝛼)¹₀𝑓_𝑛−2(𝑢)(d𝑢)^𝛼⋅1Γ(1+𝛼)¹₀[Γ(1+2𝛼)]³𝑥^3𝛼[Γ(1+𝛼)]³𝑓_𝑛−2(𝑥)^𝑛−3_𝑖=1𝑓_𝑖(𝑥)(d𝑥)^𝛼=≧⋯≧⋯_𝑛𝑖=11Γ(1+𝛼)¹₀𝑓_𝑖(𝑢)(d𝑢)^𝛼⋅1Γ(1+𝛼)¹₀[Γ(1+2𝛼)]^𝑛𝑥^𝑛𝛼[Γ(1+𝛼)]^𝑛(d𝑥)^𝛼=^𝑛_𝑖=11Γ(1+𝛼)¹₀𝑓_𝑖(𝑢)(d𝑢)^𝛼×[Γ(1+2𝛼)]^𝑛[Γ(1+𝛼)]^𝑛⋅1Γ(1+𝛼)¹₀𝑥^𝑛𝛼(d𝑥)^𝛼. (37) which yields 1Γ(1+𝛼)¹₀^𝑛_𝑖=1𝑓_𝑖(𝑥)(d𝑥)^𝛼≥[Γ(1+2𝛼)]^𝑛[Γ(1+𝛼)]^𝑛1Γ(1+𝛼)¹₀𝑥^𝑛𝛼(d𝑥)^𝛼×^𝑛_𝑖=11Γ(1+𝛼)¹₀𝑓_𝑖(𝑢)(d𝑢)^𝛼. (38) 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).

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).