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

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 𝑝>1, 1/𝑝+1/𝑞=1, 𝑓(𝑥), and 𝑔(𝑥) be continuous real-valued functions on [𝑎,𝑏]. Then, the famous Hölders inequality reads as ^𝑏_𝑎𝑓(𝑥)𝑔(𝑥)𝑑𝑥≤^𝑏_𝑎𝑓(𝑥)^𝑝𝑑𝑥^1/𝑝𝑏_𝑎𝑔(𝑥)^𝑞𝑑𝑥^1/𝑞. (1) 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 [2–11]. 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 𝑓(𝑥), 𝑔(𝑥)∈𝐶_𝛼(𝑎,𝑏), 𝑝>1, 1/𝑝+1/𝑞=1. Then, 1Γ(1+𝛼)^𝑏_𝑎𝑓(𝑥)𝑔(𝑥)(𝑑𝑥)^𝛼≤1Γ(1+𝛼)^𝑏_𝑎𝑓(𝑥)^𝑝(𝑑𝑥)^𝛼1/𝑝×1Γ(1+𝛼)^𝑏_𝑎𝑔(𝑥)^𝑞(𝑑𝑥)^𝛼1/𝑞. (2)

Recently, the local fractional calculus has attracted a lot of interest for scientists and engineers. Local fractional derivative had been introduced in [12–36]; that is the local fractional derivative was structured in [12–18, 26, 30–36], Jumarie gave the modified Riemann-Liouville derivative in [19, 20], the fractal derivative was considered in [21–25, 27–29], 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 [12–14]).

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 𝐹 is a subset of the real line and is a fractal. Let 𝑓∶(𝐹,𝑑)→(Ω^′,𝑑^′) be a bi-Lipschitz mapping. Then, there exist two positive constants 𝜌,𝜏, and 𝐹⊂𝑅, 𝜌^𝑠𝐻^𝑠(𝐹)≤𝐻^𝑠𝑓(𝐹)≤𝜏^𝑠𝐻^𝑠(𝐹), (3) such that for all 𝑥₁,𝑥₂∈𝐹, 𝜌^𝛼𝑥₁−𝑥₂^𝛼≤𝑓𝑥₁−𝑓𝑥₂≤𝜏^𝛼𝑥₁−𝑥₂^𝛼. (4)

From Lemma 1, we obtain easily 𝑓𝑥₁−𝑓𝑥₂≤𝜏^𝛼𝑥₁−𝑥₂^𝛼 (5) such that 𝑓𝑥₁−𝑓𝑥₂≤𝜀^𝛼, (6) where 𝛼 is fractal dimension of 𝐹. The result that is directly deduced from fractal geometry is related to fractal coarse-grained mass function 𝛾^𝛼[𝐹,𝑎,𝑏] which reads 𝛾^𝛼[]=𝐻𝐹,𝑎,𝑏^𝛼(𝐹∩(𝑎,𝑏))Γ(1+𝛼) (7) with 𝐻^𝛼(𝐹∩(𝑎,𝑏))=(𝑏−𝑎)^𝛼, (8) where 𝐻^𝛼 is 𝛼 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 [12–14].

Definition 2 (see [12, 14]). If there exists 𝑓(𝑥)−𝑓𝑥₀≤𝜀^𝛼, (9) with |𝑥−𝑥₀|^𝛼≤𝛿, for 𝜀,𝛿>0 and 𝜀,𝛿∈𝑅, then 𝑓(𝑥) is called local fractional continuous at 𝑥=𝑥₀, denoted by lim_𝑥→𝑥0𝑓(𝑥)=𝑓(𝑥₀). 𝑓(𝑥) is local fractional continuous on the interval (𝑎,𝑏), denoted by 𝑓(𝑥)∈𝐶_𝛼(𝑎,𝑏) (10) if (9) holds for 𝑥∈(𝑎,𝑏).

Definition 3 (see [13, 14]). Assume that 𝑓(𝑥) is a nondifferentiable function of exponent 𝛼, 0<𝛼≤1, which satisfies Hölder function of exponent 𝛼, then, for 𝑥,𝑦∈𝑋 such that 𝑓(𝑥)−𝑓𝑦≤𝐶𝑥−𝑦^𝛼. (11)

Definition 4 (see [13, 14]). A function 𝑓(𝑥) is continuous of order 𝛼, 0<𝛼≤1, or shortly 𝛼 continuous, if 𝑓(𝑥)−𝑓𝑥₀≤𝑜𝑥−𝑥₀^𝛼. (12)

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 [12–14]). Let 𝑓(𝑥)∈𝐶_𝛼(𝑎,𝑏). Local fractional derivative of 𝑓(𝑥) of order 𝛼 at 𝑥=𝑥₀ is given by 𝑓^(𝛼)𝑥₀=𝑑^𝛼𝑓(𝑥)𝑑𝑥^𝛼_𝑥=𝑥0=lim_𝑥→𝑥0Δ^𝛼𝑓(𝑥)−𝑓𝑥₀𝑥−𝑥₀^𝛼, (13) where Δ^𝛼(𝑓(𝑥)−𝑓(𝑥₀))≅Γ(1+𝛼)Δ(𝑓(𝑥)−𝑓(𝑥₀)).

For any 𝑥∈(𝑎,𝑏), there exists 𝑓^(𝛼)(𝑥)=𝐷_𝑥^(𝛼)𝑓(𝑥) (14) denoted by 𝑓(𝑥)∈𝐷_𝑥^(𝛼)(𝑎,𝑏). (15) Local fractional derivative of high order is derived as 𝑓^(𝑘𝛼)(𝑥)=𝐷_𝑥^(𝛼)⋯𝐷^𝑘times_𝑥^(𝛼)𝑓(𝑥), (16) and local fractional partial derivative of high order is derived as 𝜕^𝑘𝛼𝑓(𝑥)𝜕𝑥^𝑘𝛼=^𝑘times𝜕^𝛼𝜕𝑥^𝛼⋯𝜕^𝛼𝜕𝑥^𝛼𝑓(𝑥). (17)

Definition 7 (see [12–14]). Let 𝑓(𝑥)∈𝐶_𝛼(𝑎,𝑏). Local fractional integral of 𝑓(𝑥) of order 𝛼 in the interval [𝑎,𝑏] is deduced 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_𝑗, Δ𝑡=max{Δ𝑡₁,Δ𝑡₂,…,Δ𝑡_𝑗,…}, and [𝑡_𝑗,𝑡_𝑗+1], 𝑗=1,2,…,𝑁−1, 𝑡₀=𝑎, 𝑡_𝑁=𝑏 is a partition of the interval [𝑎,𝑏].

For convenience, we assume that _𝛼𝐼_𝛼^(𝛼)𝑓(𝑥)=0if𝑎=𝑏,_𝛼𝐼_𝑏^(𝛼)𝑓(𝑥)=−_𝑏𝐼_𝛼^(𝛼)𝑓(𝑥)if𝑎>𝑏. (19) For any 𝑥∈(𝑎,𝑏), we can get _𝛼𝐼_𝑥^(𝛼)𝑓(𝑥) (20) denoted by 𝑓(𝑥)∈_𝛼𝐼_𝑥^(𝛼)(𝑎,𝑏). (21)

Remark 8. If 𝑓(𝑥)∈𝐷_𝑥^(𝛼)(𝑎,𝑏), or _𝛼𝐼_𝑥^(𝛼)(𝑎,𝑏) then we have 𝑓(𝑥)∈𝐶_𝛼(𝑎,𝑏). (22)

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 𝑓(𝑥), 𝑔(𝑥)∈𝐶_𝛼(𝑎,𝑏), and let 0<𝑝<1, 1/𝑝+1/𝑞=1. Then, 1Γ(1+𝛼)^𝑏_𝑎𝑓(𝑥)𝑔(𝑥)(𝑑𝑥)^𝛼≥1Γ(1+𝛼)^𝑏_𝑎𝑓(𝑥)^𝑝(𝑑𝑥)^𝛼1/𝑝×1Γ(1+𝛼)^𝑏_𝑎𝑔(𝑥)^𝑞(𝑑𝑥)^𝛼1/𝑞. (23)

Proof. Set 𝑐=1/𝑝, 𝑞=−𝑝𝑑, and then we have 𝑑=𝑐/(𝑐−1). By inequality (2), we obtain 1Γ(1+𝛼)^𝑏_𝑎𝑓(𝑥)^𝑝(𝑑𝑥)^𝛼=1Γ(1+𝛼)^𝑏_𝑎𝑓(𝑥)𝑔(𝑥)^𝑝𝑔(𝑥)^−𝑝(𝑑𝑥)^𝛼≤1Γ(1+𝛼)^𝑏_𝑎𝑓(𝑥)𝑔(𝑥)^𝑝𝑐(𝑑𝑥)^𝛼1/𝑐×1Γ(1+𝛼)^𝑏_𝑎𝑔(𝑥)^−𝑝𝑑(𝑑𝑥)^𝛼1/𝑑=1Γ(1+𝛼)^𝑏_𝑎𝑓(𝑥)𝑔(𝑥)(𝑑𝑥)^𝛼1/𝑐×1Γ(1+𝛼)^𝑏_𝑎𝑔(𝑥)^𝑞(𝑑𝑥)^𝛼1−𝑝. (24) In (24), multiplying both sides by ((1/Γ(1+𝛼))∫^𝑏_𝑎|𝑔(𝑥)|^𝑞(𝑑𝑥)^𝛼)^𝑝−1 yields 1Γ(1+𝛼)^𝑏_𝑎𝑓(𝑥)^𝑝(𝑑𝑥)^𝛼1Γ(1+𝛼)^𝑏_𝑎𝑔(𝑥)^𝑞(𝑑𝑥)^𝛼𝑝−1≤1Γ(1+𝛼)^𝑏_𝑎𝑓(𝑥)𝑔(𝑥)(𝑑𝑥)^𝛼𝑝. (25) Inequality (25) implies 1Γ(1+𝛼)^𝑏_𝑎𝑓(𝑥)𝑔(𝑥)(𝑑𝑥)^𝛼≥1Γ(1+𝛼)^𝑏_𝑎𝑓(𝑥)^𝑝(𝑑𝑥)^𝛼1/𝑝×1Γ(1+𝛼)^𝑏_𝑎𝑔(𝑥)^𝑞(𝑑𝑥)^𝛼1/𝑞. (26)

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

Corollary 10. Let 𝑓_𝑗(𝑥)∈𝐶_𝛼(𝑎,𝑏), let 𝑝_𝑗∈𝑅, 𝑗=1,2,…𝑚, and let ∑^𝑚_𝑗=11/𝑝_𝑗=1. Then,(1)for 𝑝_𝑗>1, one has 1Γ(1+𝛼)^𝑏_𝑎^𝑚_𝑗=1𝑓_𝑗(𝑥)(𝑑𝑥)^𝛼≤^𝑚_𝑗=1^𝑏_𝑎1Γ()𝑓1+𝛼^𝑗(𝑥)^𝑝𝑗(𝑑𝑥)^{𝛼1/𝑝𝑗}, (27)(2)for 0<𝑝₁<1, 𝑝_𝑗<0, 𝑗=2,…,𝑚, one has 1Γ(1+𝛼)^𝑏_𝑎^𝑚_𝑗=1𝑓_𝑗(𝑥)(𝑑𝑥)^𝛼≥^𝑚_𝑗=1^𝑏_𝑎1Γ()𝑓1+𝛼^𝑗(𝑥)^𝑝𝑗(𝑑𝑥)^{𝛼1/𝑝𝑗}. (28)

Proof. (1) If 𝑚=2, 𝑝₁, 𝑝₂ are two positive constants and 1/𝑝₁+1/𝑝₂=1. In particular, setting 𝑝₁>1, 𝑝₂>1, inequality (27) becomes inequality (2). Suppose (27) holds when 𝑚≥2. Using mathematical induction, let 𝑝₁,𝑝₂,…,𝑝_𝑚+1>0 be real numbers with ∑^𝑚+1_𝑗=11/𝑝_𝑗=1 and 𝑓_𝑗(𝑥)≥0, 𝑗=1,2,…,𝑚,𝑚+1; we must have 𝑝_𝑖>1 for 𝑗=1,2,…,𝑚,𝑚+1. In particular, we have 𝑝₁𝑝>0,¹𝑝₁1−1>0,𝑝₁+𝑝₁−1𝑝₁=1. (29) By using Hölder’s inequality (2), we obtain 1Γ(1+𝛼)^𝑏_𝑎^𝑚+1_𝑗=1𝑓_𝑗(𝑥)(𝑑𝑥)^𝛼=1Γ(1+𝛼)^𝑏_𝑎𝑓₁(𝑥)^𝑚+1_𝑗=2𝑓_𝑗(𝑥)(𝑑𝑥)^𝛼≤1Γ(1+𝛼)^𝑏_𝑎𝑓₁(𝑥)^𝑝1(𝑑𝑥)^𝛼1/𝑝1×1Γ(1+𝛼)^𝑏_𝑎^𝑚+1_𝑗=2𝑓_𝑗(𝑥)^{𝑝1/(𝑝1−1)}(𝑑𝑥)^{𝛼(𝑝1−1)/𝑝1}=1Γ(1+𝛼)^𝑏_𝑎𝑓₁(𝑥)^𝑝1(𝑑𝑥)^𝛼1/𝑝1×1Γ(1+𝛼)^𝑏_𝑎^𝑚+1_𝑗=2𝑓_𝑗(𝑥)^{𝑝1/(𝑝1−1)}(𝑑𝑥)^{𝛼(𝑝1−1)/𝑝1} (30) since 𝑝_𝑗𝑝₁−1𝑝₁>0for𝑗=2,…,𝑚,𝑚+1,_𝑚+1𝑗=2𝑝₁𝑝_𝑗𝑝₁−1=𝑝₁𝑝₁−1^𝑚+1_𝑗=21𝑝_𝑗=𝑝₁𝑝₁1−11−𝑝₁=1. (31) Using induction hypothesis and inequality (30), we can get 1Γ(1+𝛼)^𝑏_𝑎^𝑚+1_𝑗=1𝑓_𝑗(𝑥)(𝑑𝑥)^𝛼≤1Γ(1+𝛼)^𝑏_𝑎𝑓₁(𝑥)^𝑝1(𝑑𝑥)^𝛼1/𝑝1×^𝑚+1_𝑗=21Γ(1+𝛼)^𝑏_𝑎𝑓_𝑗(𝑥)^{𝑝1/(𝑝1−1)⋅𝑝𝑗(𝑝1−1)/𝑝1}×(𝑑𝑥)^{𝛼𝑝1/𝑝𝑗(𝑝1−1)(𝑝1−1)/𝑝1}=1Γ(1+𝛼)^𝑏_𝑎𝑓₁(𝑥)^𝑝1(𝑑𝑥)^𝛼1/𝑝1×^𝑚+1_𝑗=21Γ(1+𝛼)^𝑏_𝑎𝑓_𝑗(𝑥)^𝑝𝑗(𝑑𝑥)^{𝛼1/𝑝𝑗}. (32) Hence, this completes the proof.

(2) The Proof of (28) is similar to the proof of (27). Clearly when 𝑚=2, inequality (28) becomes Hölder’s inequality (23). Now, suppose that (28) holds for some integer 𝑚≥2. Let 𝑝₁,𝑝₂,…,𝑝_𝑚<0, and let 𝑝_𝑚+1∈𝑅 be such that ∑^𝑚+1_𝑗=11/𝑝_𝑗=1, and let 𝑓_𝑗(𝑥)∈𝐿_𝑝,𝛼[𝑎,𝑏], 𝑗=1,2,…,𝑚,𝑚+1. Note that 0<𝑝_𝑚+1<1, since 𝑝₁𝑝>0,0<¹𝑝₁1−1<1,𝑝₁+𝑝₁−1𝑝₁=1. (33)

Observing Höder’s inequality (23), we have 1Γ(1+𝛼)^𝑏_𝑎^𝑚_𝑗=1𝑓_𝑗(𝑥)(𝑑𝑥)^𝛼=1Γ(1+𝛼)^𝑏_𝑎𝑓₁(𝑥)^𝑚_𝑗=2𝑓_𝑗(𝑥)(𝑑𝑥)^𝛼≥1Γ(1+𝛼)^𝑏_𝑎𝑓₁(𝑥)^𝑝1(𝑑𝑥)^𝛼1/𝑝1×1Γ(1+𝛼)^𝑏_𝑎^𝑚_𝑗=2𝑓_𝑗(𝑥)^{𝑝1/(𝑝1−1)}(𝑑𝑥)^{𝛼(𝑝1−1)/𝑝1}=1Γ(1+𝛼)^𝑏_𝑎𝑓₁(𝑥)^𝑝1(𝑑𝑥)^𝛼1/𝑝1×1Γ(1+𝛼)^𝑏_𝑎^𝑚_𝑗=2𝑓_𝑗(𝑥)^{𝑝1/(𝑝1−1)}(𝑑𝑥)^{𝛼(𝑝1−1)/𝑝1} (34)

unless (1/Γ(1+𝛼))∫^𝑏_𝑎|𝑓₁(𝑥)|^𝑝1(𝑑𝑥)^𝛼=0.

Since 𝑝_𝑗𝑝₁−1𝑝₁𝑝<0for𝑗=2,…,𝑚,_𝑚+1𝑝₁−1𝑝₁>0,_𝑚+1𝑗=2𝑝₁𝑝_𝑗𝑝₁−1=1. (35) Combining induction hypothesis and (34), we obtain 1Γ(1+𝛼)^𝑏_𝑎^𝑚+1_𝑗=1𝑓_𝑗(𝑥)(𝑑𝑥)^𝛼≥1Γ(1+𝛼)^𝑏_𝑎𝑓₁(𝑥)^𝑝1(𝑑𝑥)^𝛼1/𝑝1×^𝑚+1_𝑗=21Γ(1+𝛼)^𝑏_𝑎𝑓_𝑗(𝑥)^{𝑝1/(𝑝1−1)⋅𝑝𝑗(𝑝1−1)/𝑝1}×(𝑑𝑥)^{𝛼𝑝1/𝑝𝑗(𝑝1−1)(𝑝1−1)/𝑝1}=1Γ(1+𝛼)^𝑏_𝑎𝑓₁(𝑥)^𝑝1(𝑑𝑥)^𝛼1/𝑝1×^𝑚+1_𝑗=21Γ(1+𝛼)^𝑏_𝑎𝑓_𝑗(𝑥)^𝑝𝑗(𝑑𝑥)^{𝛼1/𝑝𝑗}=^𝑚+1_𝑗=11Γ(1+𝛼)^𝑏_𝑎𝑓_𝑗(𝑥)^𝑝𝑗(𝑑𝑥)^{𝛼1/𝑝𝑗} (36) unless (1/Γ(1+𝛼))∫^𝑏_𝑎|𝑓_𝑗(𝑥)|^𝑝𝑗(𝑑𝑥)^𝛼=0 for some 𝑗=1,2,…,𝑚.

4. Some Related Results

Theorem 11 (Minkowski inequality see [12]). Let 𝑓(𝑥), 𝑔(𝑥)∈𝐶_𝛼(𝑎,𝑏), 𝑝>1. Then, 1Γ(1+𝛼)^𝑏_𝑎𝑓(𝑥)+𝑔(𝑥)^𝑝(𝑑𝑥)^𝛼1/𝑝≤1Γ(1+𝛼)^𝑏_𝑎𝑓(𝑥)^𝑝(𝑑𝑥)^𝛼1/𝑝+1Γ(1+𝛼)^𝑏_𝑎𝑔(𝑥)^𝑞(𝑑𝑥)^𝛼1/𝑞. (37)

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

Theorem 12 (reverse Minkowski's inequality). Let 𝑓(𝑥), 𝑔(𝑥)∈𝐶_𝛼(𝑎,𝑏), 0<𝑝<1. Then, 1Γ(1+𝛼)^𝑏_𝑎𝑓(𝑥)+𝑔(𝑥)^𝑝(𝑑𝑥)^𝛼1/𝑝≥1Γ(1+𝛼)^𝑏_𝑎𝑓(𝑥)^𝑝(𝑑𝑥)^𝛼1/𝑝+1Γ(1+𝛼)^𝑏_𝑎𝑔(𝑥)^𝑞(𝑑𝑥)^𝛼1/𝑞. (38)

Proof. Let 1𝑀=Γ(1+𝛼)^𝑏_𝑎𝑓(𝑥)^𝑝(𝑑𝑥)^𝛼,1𝑁=Γ(1+𝛼)^𝑏_𝑎𝑔(𝑥)^𝑞(𝑑𝑥)^𝛼,1𝑊=Γ(1+𝛼)^𝑏_𝑎𝑓(𝑥)^𝑝(𝑑𝑥)^𝛼1/𝑝+1Γ(1+𝛼)^𝑏_𝑎𝑔(𝑥)^𝑞(𝑑𝑥)^𝛼1/𝑞. (39) By the Hölder inequality, in view of 0<𝑝<1, we have 1𝑊=Γ(1+𝛼)×^𝑏_𝑎𝑓(𝑥)^𝑝𝑀^1/𝑝−1+𝑔(𝑥)^𝑝𝑁^1/𝑝−1(𝑑𝑥)^𝛼≤1Γ(1+𝛼)^𝑏_𝑎𝑓(𝑥)+𝑔(𝑥)^𝑝𝑀^1/𝑝+𝑁^{1/𝑝1−𝑝}(𝑑𝑥)^𝛼=𝑊^1−𝑝1Γ(1+𝛼)^𝑏_𝑎𝑓(𝑥)+𝑔(𝑥)^𝑝(𝑑𝑥)^𝛼. (40) By inequality (40), reverse Minkowski's inequality and the theorem are completely proved.

Corollary 13. Let 𝑓_𝑗(𝑥)∈𝐶_𝛼(𝑎,𝑏), 𝑗=1,2,…,𝑚.(1)For 𝑝>1, one has 1Γ(1+𝛼)^𝑏_𝑎^𝑚_𝑗=1𝑓_𝑗(𝑥)^𝑝(𝑑𝑥)^𝛼1/𝑝≤^𝑚_𝑗=11Γ(1+𝛼)^𝑏_𝑎𝑓(𝑥)^𝑝(𝑑𝑥)^𝛼1/𝑝. (41)(2)For 0<𝑝<1, one has 1Γ(1+𝛼)^𝑏_𝑎^𝑚_𝑗=1𝑓_𝑗(𝑥)^𝑝(𝑑𝑥)^𝛼1/𝑝≥^𝑚_𝑗=11Γ(1+𝛼)^𝑏_𝑎𝑓(𝑥)^𝑝(𝑑𝑥)^𝛼1/𝑝. (42)

Proof. (1) Using Theorem 11, we have 1Γ(1+𝛼)^𝑏_𝑎^𝑚_𝑗=1𝑓_𝑗(𝑥)^𝑝(𝑑𝑥)^𝛼≤1Γ(1+𝛼)^𝑏_𝑎𝑓₁(𝑥)^𝑚_𝑗=1𝑓_𝑗(𝑥)^𝑝−1(𝑑𝑥)^𝛼1+⋯+Γ(1+𝛼)^𝑏_𝑎𝑓_𝑚(𝑥)^𝑚_𝑗=1𝑓_𝑗(𝑥)^𝑝−1(𝑑𝑥)^𝛼≤1Γ(1+𝛼)^𝑏_𝑎𝑓₁(𝑥)^𝑝(𝑑𝑥)^𝛼1/𝑝×1Γ(1+𝛼)^𝑏_𝑎^𝑚_𝑗=1𝑓_𝑗(𝑥)^{𝑞(𝑝−1)}(𝑑𝑥)^𝛼1/𝑞1+⋯+Γ(1+𝛼)^𝑏_𝑎𝑓_𝑚(𝑥)^𝑝(𝑑𝑥)^𝛼1/𝑝×1Γ(1+𝛼)^𝑏_𝑎^𝑚_𝑗=1𝑓_𝑗(𝑥)^{𝑞(𝑝−1)}(𝑑𝑥)^𝛼1/𝑞=^𝑚_𝑗=11Γ(1+𝛼)^𝑏_𝑎𝑓_𝑗(𝑥)^𝑝(𝑑𝑥)^𝛼1/𝑝×1Γ(1+𝛼)^𝑏_𝑎^𝑚_𝑗=1𝑓_𝑗(𝑥)^𝑝(𝑑𝑥)^𝛼1/𝑞. (43) Multiplying ((1/Γ(1+𝛼))∫^𝑏_𝑎|∑^𝑚_𝑗=1𝑓_𝑗(𝑥)|^𝑝(𝑑𝑥)^𝛼)^1/𝑞 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 𝑓_𝑗(𝑥)∈𝐶_𝛼(𝑎,𝑏), 𝑗=1,2,…,𝑚. Then,(1)for 𝑝>1, one has 1Γ(1+𝛼)^𝑏_𝑎^𝑚_𝑗=1𝑓_𝑗(𝑥)^𝑝(𝑑𝑥)^𝛼>^𝑚_𝑗=11Γ(1+𝛼)^𝑏_𝑎𝑓_𝑗(𝑥)^𝑝(𝑑𝑥)^𝛼, (44)(2)for 0<𝑝<1, one has 1Γ(1+𝛼)^𝑏_𝑎^𝑚_𝑗=1𝑓_𝑗(𝑥)^𝑝(𝑑𝑥)^𝛼<^𝑚_𝑗=11Γ(1+𝛼)^𝑏_𝑎𝑓_𝑗(𝑥)^𝑝(𝑑𝑥)^𝛼. (45)

Theorem 15 (Dresher’s inequality). Let 𝑓(𝑥), 𝑔(𝑥)∈𝐶_𝛼(𝑎,𝑏), and let 0<𝑟<1<𝑝; then (1/Γ(1+𝛼))∫^𝑏_𝑎𝑓(𝑥)+𝑔(𝑥)^𝑝(𝑑𝑥)^𝛼(1/Γ(1+𝛼))∫^𝑏_𝑎𝑓(𝑥)+𝑔(𝑥)^𝑟(𝑑𝑥)^{𝛼1/(𝑝−𝑟)}≤(1/Γ(1+𝛼))∫^𝑏_𝑎𝑓(𝑥)^𝑝(𝑑𝑥)^𝛼(1/Γ(1+𝛼))∫^𝑏_𝑎𝑓(𝑥)^𝑟(𝑑𝑥)^{𝛼1/(𝑝−𝑟)}+(1/Γ(1+𝛼))∫^𝑏_𝑎𝑔(𝑥)^𝑝(𝑑𝑥)^𝛼(1/Γ(1+𝛼))∫^𝑏_𝑎𝑔(𝑥)^𝑟(𝑑𝑥)^{𝛼1/(𝑝−𝑟)}. (46)

Proof. Combining inequality (2) and Theorem 11, we have 1Γ(1+𝛼)^𝑏_𝑎𝑓(𝑥)+𝑔(𝑥)^𝑝(𝑑𝑥)^{𝛼1/(𝑝−𝑟)}≤1Γ(1+𝛼)^𝑏_𝑎𝑓(𝑥)^𝑝(𝑑𝑥)^𝛼1/𝑝+1Γ(1+𝛼)^𝑏_𝑎𝑔(𝑥)^𝑝(𝑑𝑥)^{𝛼1/𝑝𝑝/(𝑝−𝑟)}=∫^𝑏_𝑎𝑓(𝑥)^𝑝(𝑑𝑥)^𝛼∫^𝑏_𝑎𝑓(𝑥)^𝑟(𝑑𝑥)^𝛼1/𝑝×1Γ(1+𝛼)^𝑏_𝑎𝑓(𝑥)^𝑟(𝑑𝑥)^𝛼1/𝑝+∫^𝑏_𝑎𝑔(𝑥)^𝑝(𝑑𝑥)^𝛼∫^𝑏_𝑎𝑔(𝑥)^𝑟(𝑑𝑥)^𝛼1/𝑝×1Γ(1+𝛼)^𝑏_𝑎𝑔(𝑥)^𝑟(𝑑𝑥)^{𝛼1/𝑝𝑝/(𝑝−𝑟)}≤(1/Γ(1+𝛼))∫^𝑏_𝑎𝑓(𝑥)^𝑝(𝑑𝑥)^𝛼(1/Γ(1+𝛼))∫^𝑏_𝑎𝑓(𝑥)^𝑟(𝑑𝑥)^{𝛼1/(𝑝−𝑟)}+(1/Γ(1+𝛼))∫^𝑏_𝑎𝑔(𝑥)^𝑝(𝑑𝑥)^𝛼(1/Γ(1+𝛼))∫^𝑏_𝑎𝑔(𝑥)^𝑟(𝑑𝑥)^{𝛼1/(𝑝−𝑟)}×1Γ(1+𝛼)^𝑏_𝑎𝑓(𝑥)^𝑟(𝑑𝑥)^𝛼1/𝑟+1Γ(1+𝛼)^𝑏_𝑎𝑔(𝑥)^𝑟(𝑑𝑥)^{𝛼1/𝑟𝑟/(𝑝−𝑟)}. (47) Using reverse Minkowski inequality (38), we have 1Γ(1+𝛼)^𝑏_𝑎𝑓(𝑥)^𝑟(𝑑𝑥)^𝛼1/𝑟+1Γ(1+𝛼)^𝑏_𝑎𝑔(𝑥)^𝑟(𝑑𝑥)^{𝛼1/𝑟𝑟}≤1Γ(1+𝛼)^𝑏_𝑎𝑓(𝑥)+𝑔(𝑥)^𝑟(𝑑𝑥)^𝛼. (48) By (47) and (48), we deduce that (46) holds. This completes the proof of the theorem.

Corollary 16. Let 𝑓_𝑗(𝑥)∈𝐶_𝛼(𝑎,𝑏), 0<𝑟<1<𝑝, 𝑗=1,2,…,𝑚. Then, (1/Γ(1+𝛼))∫^𝑏_𝑎∑^𝑚_𝑗=1𝑓_𝑗(𝑥)^𝑝(𝑑𝑥)^𝛼(1/Γ(1+𝛼))∫^𝑏_𝑎∑^𝑚_𝑗=1𝑓_𝑗(𝑥)^𝑟(𝑑𝑥)^{𝛼1/(𝑝−𝑟)}≤^𝑚_𝑗=1(1/Γ(1+𝛼))∫^𝑏_𝑎𝑓_𝑗(𝑥)^𝑝(𝑑𝑥)^𝛼(1/Γ(1+𝛼))∫^𝑏_𝑎𝑓_𝑗(𝑥)^𝑟(𝑑𝑥)^{𝛼1/(𝑝−𝑟)}. (49)

Theorem 17 (reverse Dresher’s inequality). Let 𝑓(𝑥), 𝑔(𝑥)∈𝐶_𝛼(𝑎,𝑏), 𝑝≤0≤𝑟<1. Then, (1/Γ(1+𝛼))∫^𝑏_𝑎𝑓(𝑥)+𝑔(𝑥)^𝑝(𝑑𝑥)^𝛼(1/Γ(1+𝛼))∫^𝑏_𝑎𝑓(𝑥)+𝑔(𝑥)^𝑟(𝑑𝑥)^{𝛼1/(𝑝−𝑟)}≥(1/Γ(1+𝛼))∫^𝑏_𝑎𝑓(𝑥)^𝑝(𝑑𝑥)^𝛼(1/Γ(1+𝛼))∫^𝑏_𝑎𝑓(𝑥)^𝑟(𝑑𝑥)^{𝛼1/(𝑝−𝑟)}+(1/Γ(1+𝛼))∫^𝑏_𝑎𝑔(𝑥)^𝑝(𝑑𝑥)^𝛼(1/Γ(1+𝛼))∫^𝑏_𝑎𝑔(𝑥)^𝑟(𝑑𝑥)^{𝛼1/(𝑝−𝑟)}. (50)

Proof. Let 𝛼₁≥0, 𝛼₂≥0, 𝛽₁>0, 𝛽₂>0, and −1<𝜆<0, using Radon’s inequality (see [3]) ^𝑛_𝑘=1𝑎^𝑝_𝑘𝑏_𝑘^𝑝−1<∑^𝑛_𝑘=1𝑎_𝑘^𝑝∑^𝑛_𝑘=1𝑏_𝑘^𝑝−1,𝑎_𝑘≥0,𝑏_𝑘>0,0<𝑝<1. (51) We have 𝛼₁^𝜆+1𝛽^𝜆₁+𝛼₂^𝜆+1𝛽^𝜆₂≤𝛼₁+𝛼₂^𝜆+1𝛽₁+𝛽₂^𝜆, (52) if and only if sequence (𝛼) and sequence (𝛽) are proportional. Let 𝛼₁=1Γ(1+𝛼)^𝑏_𝑎𝑓(𝑥)^𝑝(𝑑𝑥)^𝛼1/𝑝,𝛽₁=1Γ(1+𝛼)^𝑏_𝑎𝑓(𝑥)^𝑟(𝑑𝑥)^𝛼1/𝑟,𝛼₂=1Γ(1+𝛼)^𝑏_𝑎𝑔(𝑥)^𝑝(𝑑𝑥)^𝛼1/𝑝,𝛽₂=1Γ(1+𝛼)^𝑏_𝑎𝑔(𝑥)^𝑟(𝑑𝑥)^𝛼1/𝑟, (53) and set 𝜆=𝑟/(𝑝−𝑟). Observing (52)–(53), we have 𝛼₁^𝜆+1𝛽^𝜆₁+𝛼₂^𝜆+1𝛽^𝜆₂=(1/Γ(1+𝛼))∫^𝑏_𝑎𝑓(𝑥)^𝑝(𝑑𝑥)^{𝛼(𝜆+1)/𝑝}(1/Γ(1+𝛼))∫^𝑏_𝑎𝑓(𝑥)^𝑟(𝑑𝑥)^{𝛼𝜆/𝑟}+(1/Γ(1+𝛼))∫^𝑏_𝑎𝑔(𝑥)^𝑝(𝑑𝑥)^{𝛼(𝜆+1)/𝑝}(1/Γ(1+𝛼))∫^𝑏_𝑎𝑔(𝑥)^𝑟(𝑑𝑥)^{𝛼𝜆/𝑟}=(1/Γ(1+𝛼))∫^𝑏_𝑎𝑓(𝑥)^𝑝(𝑑𝑥)^𝛼(1/Γ(1+𝛼))∫^𝑏_𝑎𝑓(𝑥)^𝑟(𝑑𝑥)^{𝛼1/(𝑝−𝑟)}+(1/Γ(1+𝛼))∫^𝑏_𝑎𝑔(𝑥)^𝑝(𝑑𝑥)^𝛼(1/Γ(1+𝛼))∫^𝑏_𝑎𝑔(𝑥)^𝑟(𝑑𝑥)^{𝛼1/(𝑝−𝑟)}≤𝛼₁+𝛼₂^𝜆+1𝛽₁+𝛽₂^𝜆=1Γ(1+𝛼)^𝑏_𝑎𝑓(𝑥)^𝑝(𝑑𝑥)^𝛼1/𝑝+1Γ(1+𝛼)^𝑏_𝑎𝑔(𝑥)^𝑝(𝑑𝑥)^{𝛼1/𝑝𝑝/(𝑝−𝑟)}×1Γ(1+𝛼)^𝑏_𝑎𝑓(𝑥)^𝑟(𝑑𝑥)^𝛼1/𝑟+1Γ(1+𝛼)^𝑏_𝑎𝑔(𝑥)^𝑟(𝑑𝑥)^{𝛼1/𝑟𝑟/(𝑝−𝑟)−1}. (54) Since −1<𝜆=𝑟/(𝑝−𝑟)<0, let 𝑝<0<𝑟, and let 0<𝑟≤1, and by Theorem 12, we obtain, respectively, 1Γ(1+𝛼)^𝑏_𝑎𝑓(𝑥)^𝑝(𝑑𝑥)^𝛼1/𝑝+1Γ(1+𝛼)^𝑏_𝑎𝑔(𝑥)^𝑝(𝑑𝑥)^{𝛼1/𝑝𝑝}≥1Γ(1+𝛼)^𝑏_𝑎𝑓(𝑥)+𝑔(𝑥)^𝑝(𝑑𝑥)^𝛼,1Γ(1+𝛼)^𝑏_𝑎𝑓(𝑥)^𝑟(𝑑𝑥)^𝛼1/𝑟+1Γ(1+𝛼)^𝑏_𝑎𝑔(𝑥)^𝑟(𝑑𝑥)^{𝛼1/𝑟𝑟}≤1Γ(1+𝛼)^𝑏_𝑎𝑓(𝑥)+𝑔(𝑥)^𝑟(𝑑𝑥)^𝛼. (55)
Observing (54)-(55), we obtain the desired results, and the theorem is completely proved.

Corollary 18. Let 𝑓_𝑗(𝑥)∈𝐶_𝛼(𝑎,𝑏), and let 𝑝≤0≤𝑟<1, 𝑗=1,2,…,𝑚. Then, (1/Γ(1+𝛼))∫^𝑏_𝑎∑^𝑚_𝑗=1𝑓_𝑗(𝑥)^𝑝(𝑑𝑥)^𝛼(1/Γ(1+𝛼))∫^𝑏_𝑎∑^𝑚_𝑗=1𝑓_𝑗(𝑥)^𝑟(𝑑𝑥)^{𝛼1/(𝑝−𝑟)}≥^𝑚_𝑗=1(1/Γ(1+𝛼))∫^𝑏_𝑎𝑓_𝑗(𝑥)^𝑝(𝑑𝑥)^𝛼(1/Γ(1+𝛼))∫^𝑏_𝑎𝑓_𝑗(𝑥)^𝑟(𝑑𝑥)^{𝛼1/(𝑝−𝑟)}. (56)

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).