# The Improvement of Hölder’s Inequality withr-Conjugate Exponents Relative to the Lp-Norm

Authors: Xiaojun Liang; Aying Wan
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

This paper investigates Hölder’s inequality under the condition ofr-conjugate exponents in the sense that ∑k=1s1/pk=1/r. Successively, we have, under r-conjugate exponents relative to the Lp-norm, investigated generalized Hölder’s inequality, the interpolation of Hölder’s inequality, and generalized s-order Hölder’s inequality which is an expansion of the known Hölder’s inequality.


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## Body

## 1. Introduction

The celebrated Hölder inequality is one of the most important inequalities in mathematics and statistics. It is applied widely in dealing with many problems from social science, management science, and natural science. Its classical version for integrals is of the following formulation: for any measurable functionsf and g on some measurable space Ω,A,μ, it follows that(1)fg1≤fpgq,for all p,q>0 with 1/p+1/q=1, where fp=μfp1/p=∫fpdμ<∞, for all measurable functions f:Ω⟶ℝ (for details, see [1, 2]).Since Hölder’s inequality has been extensively investigated and applied to some new fields, many literature studies are contributed to the refinement of Hölder’s inequality according to specific applied fields. These improvements mainly incorporate in the following fields.The first refinement is relative to the higher integrability of gradients of solutions to various partial differential equations. It characterizes the following style: there existc, ε>0 only depending on n, p, and Λ so that(2)∬Qr,rp∇up+εdxdt≤c∬Q2r,2rpup2rp+∇updxdtε∬Q2r,2rp∇updxdt,for all time-space cylinders with Q2r,2rp=I2rp×B2r⊂0,T×Ω, where u≥0. The recent representative articles with respect to this topic are referred to in [3–6].The other refinements are Hölder’s inequality and inverse Hölder’s inequality for the pseudo-integral established by Agahi et al. [7]. Assume that p and q are a pair of conjugate exponents and that Ω,A,μ is a measure space. Suppose that f,g:Ω⟶a,b are two measurable functions and that a generator g:a,b⟶0,∞ of the pseudo-addition ⊕ and the pseudo-multiplication ⊙ is an increasing function. Then, for any σ-⊕-measure μ, it follows that(3)∫Ω⊕f⊙g⊙dμ≤∫Ω⊕f⊙p⊙dμ⊙1/p⊙∫Ω⊕g⊙q⊙dμ⊙1/q.The recent research studies contribute to interesting extensions of Hölder’s inequality for the decomposition integral, Sugeno integral, and pseudo-integral (for more details, see [7–12]) since fuzzy measures and Sugeno integral have been successfully applied to various fields such as to decision-making [13] and to artificial intelligence [14]. Besides, Khan et al. investigated the converses of the Jensen inequality in their articles such as [15–18].The fundamental refinement is the improvement of Hölder’s inequality itself. As far as in 1961, Beckenbach and Bellman [19] derived generalized Hölder’s inequality of the following form: let m and n be positive integers, and let aij>0,1≤i≤n,1≤j≤m, pj>0, and ∑i=1m1/pj=1. Then, it follows that(4)∑i=1n∏j=1maij≤∏j=1m∑i=1naijpj1/pj.The corresponding integral form is(5)∫ab∏j=1mfjxdx≤∏j=1m∫abfjpjxdx1/pj,where fjx>0,j=1,2,…,m, x∈a,b, −∞<a<b<∞,pj>0,∑j=1m1/pj=1, and fj∈Lpja,b. Note that, by taking m=2, f1x=fx, and f2x=gx, inequality (5) reduces to inequality (1). Qiang and Hu [20], then, derived further contributions on this topic as follows: let aij>0, pk>0, αij∈ℝ, i=1,2,…,n;j=1,2,…,m;k=1,2,…,s, ∑k=1s1/pk=1, and ∑k=1sαij=0. Then,(6)∑i=1n∏j=1maij≤∏k=1s∑i=1n∏j=1maij1+pkαkj1/pk.Moreover, for the integral form of the above inequality, iffjx>0,j=1,2,…,m,x∈a,b,−∞<a<b<∞, and fj∈Ca,b, then(7)∫ab∏j=1mfjxdx≤∏k=1s∫ab∏j=1mfj1+pkαkjx1/pk.Many existing inequalities related to the Hölder inequality are special cases of inequalities (6) and (7). For example, putting s=m,αkj=−1/pk for j≠k, and αkk=1−1/pk, inequalities (6) and (7) are reduced to (4) and (5), respectively. The condition ∑k=1s1/pk=1 may be regarded as Hölder’s inequality with a system of generalized conjugate exponents which will be introduced in Section 2. Recently, Masjed-Jamei [21] established an extension of the Callebaut inequality [22], that is,(8)∫ab∏j=1mfjxdx≤∫ab∏j=1mfj1+αjxdx1/2∫ab∏j=1mfj1−αjxdx1/2,where αj∈ℝ and fjx>0, i=1,2,…,n;j=1,2,…,m.Based on these contributions of Masjed-Jamei, Qiang and Hu [20] derived the following interesting result:(9)∫ab∏j=1mfjxdx≤ϕc≤∏k=1s∫ab∏j=1mfj1+rpkαkjxdx1/rpk,where(10)ϕc=∫ac∏j=1mfjxdx+∏k=1s∫cb∏j=1mfj1+rpkαkjxdx1/rpkis a nonincreasing function with a≤c≤b.These aforementioned literature studies have generalized Hölder’s inequality upon the condition∑k=1s1/pk=r. On the one hand, however, this condition lacks consistency in the form. On the other hand, these inequalities lack relevance on the two sides of the inequalities. Recall the mean value theorem of integrals: let fx be measurable on a,b and gx be a monotone function. Then, there exists ξ∈a,b such that(11)∫abfxgxdx=ga∫aξfxdx+gb∫ξbfxdx.Puttingm=2,s=1,r=1, and α1j=0, equation (11) will reduce to the mean value theorem of integrals. Furthermore, Tian et al. [9, 23, 24] investigated the monotone relationship between adjoining exponents of Fsn or Gstl,a≤t1≤t2≤b.In this paper, our contributions are concluded as follows. We have derived generalized Hölder’s inequality under the condition ofr-conjugate exponents in the sense that ∑k=1s1/pk=1/r. Successively, we have, under r-conjugate exponents relative to the Lp-norm, checked generalized Hölder’s inequality, the interpolation of Hölder’s inequality, and generalized s-order Hölder’s inequality which is an expansion of the known Hölder’s inequality.


## 2. Preliminary Definitions and Notations

First, we would provide some preliminary definitions for building the formulation of Hölder’s inequality withs-order r-conjugate exponents.Definition 1.
Ifp and q are a pair of positive real numbers such that p+q=pq or equivalently(12)1p+1q=1,then the positive real numbers p and q are referred to as a pair of conjugate exponents.It is clear that equation (12) implies that 1<p<∞ and 1<q<∞ and that q⟶∞ as p⟶1. Hence, it is reasonable to regard 1 and ∞ as a pair of conjugate exponents [25].Definition 2.
Ifp1,p2,…,ps are s positive real numbers such that(13)∑k=1s1pk=1,then the sequence pk,k=1,2,…,s, is known as s-order conjugate exponents.Definition 3.
Ifp and q are positive real numbers such that rp+q=pq or equivalently(14)1p+1q=1r,then the numbers p and q are referred to as a pair of r-conjugate exponents.Definition 4.
Ifp1,p2,…,ps are s positive real numbers such that(15)∑k=1s1pk=1r,then the positive real numbers pk,k=1,2,…,s, are referred to as s-order r-conjugate exponents.Given the definitions of conjugate exponents, our next aim is to define the integral and to further recall thep-norm.Definition 5.
The integral of a real-valued, measurable functionf on some measure space Ω,A,μ is defined as(16)μf=∫fdμ=∫fωμdω.Iff is simple and nonnegative, hence of the form c11A1+⋯+cn1An for some n∈Z+, A1,…,An∈A, and c1,…,cn∈ℛ+, μf is defined as(17)μf=c1μA1+c2μA2+⋯+cnμAn.For any nonnegative measurable functionf, we may choose some simple measurable functions f1,f2,… with 0≤fn↑f, and define μf=limnμfn [2].Note that it is customary to relate all objects of the study to a basic probability spaceΩ,A,μ, which is nothing more than a normalized measure space.Definition 6.
Given a measure spaceΩ,A,μ and p≥1, we write Lp=LpΩ,A,μ for the class of all measurable functions f on Ω,A,μ with(18)fp=μfp1/p=∫Ωfpdμ1/p,and call fp the Lp-norm of f and LpΩ,A,μ the Lp-space.


## 3. Main Results

### 3.1. Generalized Hölder’s Inequality with a Pair ofr-Conjugate Exponents

In this part, we will generalize the celebrated Young inequality and Hölder inequality for integrals to those withr-conjugate exponents.Theorem 1.
(generalized Young inequality). Assume thatp and q are a pair of r-conjugate exponents. Then, for positive real numbers a and b,(19)abrr≤app+bqq,with equality if and only if ap=bq.Proof.
Puttingfx=ex, we have f′x=f″x=ex>0 for any real number x∈ℝ. Thus, fx is convex for x∈0,+∞. Assuming x=plna and y=qlnb, due to Jensen’s inequality, we obtain(20)frpx+rqy≤rpfx+rqfy.
Substituting, respectively,plna and qlnb for x and y, it follows that(21)erx/p+ry/q≤rpex+rqey,that is, abr/r≤ap/p+bq/q. The equality holds if and only if rx=ry or equally, ap=bq.
Then, we will derive the generalized Hölder inequality with a pair of conjugate exponents.Theorem 2.
(Hölder’s inequality with a pair of conjugate exponents). Letp and q be a pair of conjugate exponents. For any measurable functions f and g on some measure space Ω,A,μ, we have(22)fg1≤fpgq,with equality if and only if there exist two constants λ1 and λ2 which need not vanish such that λ1fp+λ2gq=0μ a.e.Proof.
One could refer to [22] when the need arises.Corollary 1.
For every measurable functionf on some measure space Ω,A,μ, we have(23)f1≤fp,for 1<p<∞.Proof.
It is trivial to prove this conclusion by puttingg=1 in Theorem 2.
The simplest measure on a measurable spaceΩ,A is the unit masses or Dirac measure δx,x∈Ω, defined by δxA=1Ax. We may form, for any countable set A=x1,x2,…, the associated counting measure μ=∑nδxn.
Every measurable mappingf of Ω into some measurable space S,S is referred to as a random element in S. A random element in S is called a random variable when S=ℝ, a random vector when S=ℝn, and a random sequence when S=ℝ∞, respectively.
Ifμ is Lebesgue measure on ℝn, we write Lpℝn instead of LpΩ,A,μ. If μ is the counting measure on a set A, it is customary to denote the corresponding Lp-space by ℒpA. In that case, the p-norm of a random vector x=x1,x2,…,xn or a random sequence x=x1,x2,… is xp=∑k=1nxkp1/p or xp=∑k=1∞xkp1/p, respectively. Thus, the Hölder inequality for sums is similar as that for integrals.Theorem 3.
(Hölder’s inequality with a pair ofr-conjugate exponents). Let p and q be a pair of r-conjugate exponents. For any measurable functions f and g on some measure space Ω,A,μ, we have(24)fgr≤fpgq,with equality if and only if there exist two constants λ1 and λ2 which need not vanish such that λ1fp+λ2gq=0μ a.e.Proof.
By Theorem2 with r/p+r/q=1, we have(25)fg1≤fp/rgq/r.
This impliesμfg≤μfp/rr/pμgq/rr/q, and furthermore, μfg1/r≤μfp/r1/pμgq/r1/q.
Replacingf and g by fr and fr, respectively, we obtain(26)fgr=μfgr1/r≤μfp1/pμgq1/q=fpgq.
Then, the equality holds in equation (18) if and only if λ1fp/r+λ2gq/r=0. Replacing f and g by fr and gr, respectively, we obtain λ1fp+λ2gq=0. Hence, the conclusion follows.Corollary 2.
For any measurable functionf on some measure space Ω,A,μ and nonnegative real numbers p and q with p≤q, we have(27)fp≤fq.Proof.
First, note that bothr<p and r<q hold when p and q are a pair of r-conjugate exponents. Then, putting g=1 in Theorem 5, we derive fr≤fp. Therefore, it follows that fp≤gq with p≤q.
Remark: Corollary2 implies that Lp-spaces are nonincreasing with respect to p.


### 3.2. Interpolation of Hölder’s Inequality withr-Conjugate Exponents

In literature [26], İşcan derived the interpolation of Hölder’s inequality with a pair of conjugate exponents with respect to integrable functions on u,v⊂ℝ. Furthermore, Gowda [27] gave an interpolation theorem relative to spectral norms. We will extend the interpolation of Hölder’s inequality to the r-conjugate exponents’ case on measure space Ω,A,μ.Definition 7.
A nondecreasing and countably subadditive set functionμ:2Ω⟶ℝ¯+ with μ∅=0 being referred to as an outer measure on a space Ω. Furthermore, given an outer measure μ on Ω, we say that a set A⊂Ω is μ-measurable if(28)μE=μE∩A+μE∩Ac,E⊂Ω.Theorem 4.
Assumep and q to be a pair of conjugate exponents and both f and g to be measurable functions on LpΩ,A,μ. Then, for any μ-measurable set A∈A, it follows that(29)fg1≤μAμΩ1/pfpμAμΩ1/qgq+μAcμΩ1/pfpμAcμΩ1/qgq≤fpgq.Proof.
Since setA is μ-measurable, it follows that μΩ=μΩ∩A+μΩ∩Ac=μA+μAc.
We first prove the left inequality of equation (23). By Theorem 2, we may obtain(30)fg1=1μΩ∫ΩμAfgdμ+∫ΩμAcfgdμ=1μΩ∫ΩμA1/pfμA1/qgdμ+∫ΩμAc1/pfμAc1/qgdμ≤1μΩ∫ΩμAfpdμ1/p∫ΩμAgqdμ1/q+∫ΩμAcfpdμ1/p∫ΩμAcgqdμ1/q=μAμΩ1/pfpμAμΩ1/qgq+μAcμΩ1/pfpμAcμΩ1/qgq.
Hence, the left inequality of equation (23) follows.
Furthermore, iffpgq=0, then f=0μ a.e. or g=0μ a.e. Thus, we have either fp=0 and ⋅fp=0 or gq=0 and ⋅gq=0. Therefore, the right inequality of equation (23) holds, that is,(31)μAμΩ1/pfpμAμΩ1/qgq+μAcμΩ1/pfpμAcμΩ1/qgq=fpgq.
Now, we consider the case ofM=fpgq≠0. By Young’s inequality, we derive(32)1μΩMμA1/pfpμA1/qgq+μAc1/pfpμAc1/qgq=1μΩ∫ΩμAfpdμ∫Ωfpdμ1/p∫ΩμAgqdμ∫Ωgqdμ1/q+∫ΩμAcfpdμ∫Ωfpdμ1/p∫ΩμAcgqdμ∫Ωgqdμ1/q≤1μΩ∫ΩμAfpdμp∫Ωfpdμ+∫ΩμAgqdμq∫Ωgqdμ+∫ΩμAcfpdμp∫Ωfpdμ+∫ΩμAcgqdμq∫Ωgqdμ=1p+1q=1.
Hence, the right inequality of equation (23) holds, that is,(33)1μΩμA1/pfpμA1/qgq+μAc1/pfpμAc1/qgq≤fpgq.
Therefore, equation (23) follows.Theorem 5.
Assumep and q to be a pair of r-conjugate exponents and both f and g to be measurable functions on LpΩ,A,μ. Then, for any μ-measurable set A∈A, it follows that(34)fgrr≤μAμΩ1/pfprμAμΩ1/qgqr+μAcμΩ1/pfprμAcμΩ1/qgqr≤fprgqr.Proof.
Letp and q be a pair of r-conjugate exponents, i.e., r/p+r/q=1. By Theorem 6, we derive(35)fg1≤μAμΩr/pfp/rμAμΩr/qgq/r+μAcμΩr/pfp/rμAcμΩr/qgq/r≤fp/rgq/r.
It follows that(36)∫Ωfgdμ≤∫ΩμAμΩfp/rdμr/p∫ΩμAμΩgq/rdμr/q+∫ΩμAcμΩfp/rdμr/p∫ΩμAcμΩgq/rdμr/q≤∫Ωfp/rdμr/p∫Ωgq/rdμr/q.
Replacingf and g by fr and gr, respectively, it follows that(37)∫Ωfgrdμ≤∫ΩμAμΩfpdμr/p∫ΩμAμΩgqdμr/q+∫ΩμAcμΩfpdμr/p∫ΩμAcμΩgqdμr/q≤∫Ωfpdμr/p∫Ωgqdμr/q,that is,(38)fgrr≤μAμΩ1/pfprμAμΩ1/qfqr+μAcμΩ1/pfprμAcμΩ1/qfqr≤fprgqr.
This completes the proof.


### 3.3. Generalized Hölder’s Inequality withs-Order r-Conjugate Exponents

In this part, we will present the most important results of this paper in the following theorems.Theorem 6.
(Hölder’s inequality withs-order r-conjugate exponents). Assume that pk,k=1,2,…,s, are s-order r-conjugate exponents and that fjx>0,j=1,2,…,m, are all measurable functions on LpΩ,A,μ. Then,(39)∏k=1sfxkr≤∏k=1sfkxpk.Proof.
We prove this conclusion by induction.
Ifs=2, the result reduces to Hölder’s inequality with a pair of conjugate exponents.
Assume that the conclusion holds whens=l. Then, putting s=l+1, we have(40)∏k=1l+1fkxr=∏k=1lfkxfl+1xr≤∏k=1lfkxqfl+1xpl+1,where 1/p1+1/p2+⋯+1/pl=1/q.
By the aforementioned assumption, it follows that(41)∏k=1lfkxq=∫Ω∏k=1lfkxdx1/q≤∏k=1l∫Ωfkqxpk/qdxq/pk1/q=∏k=1l∫Ωfkpkx1/pk.
Thus, combining the two inequalities above, we derive(42)∏k=1l+1fkxr≤∏k=1l∫Ωfkpkxdx1/pkfl+1xpl+1=∏k=1l+1fkxpk.
This proves the cases=l+1.Theorem 7.
(generalized Hölder’s inequality withs-order r-conjugate exponents). Assume that pk,k=1,2,…,s, are s-order r-conjugate exponents, αkj∈ℝ,j=1,2,…,m, and fjx>0,j=1,2,…,m, are all measurable functions on LpΩ,A,μ. Then,(43)∏j=1mfjxr≤∏k=1s∫Ω∏j=1mfjr+pk/rαkjxdx1/pk.Proof.
Assume the positive function sequence(44)gkx=∏j=1mfjr+pk/rαkjx1/pk,k=1,2,…,s.
By simple computation and rearrangement, it follows that(45)∫Ω∏k=1sgkrxdx=∫Ω∏k=1s∏j=1mfjr+pk/rαkjr/pkdx=∫Ω∏j=1mfjrxdx,with the assumptions ∑k=1s1/pk=1/r and ∑k=1sαkj=0.
Then, by using Hölder’s inequality withs-order r-conjugate exponents, it follows that(46)∏j=1mfjxr=∏k=1sgkxr≤∏k=1sgkxpk=∏k=1s∫Ω∏j=1mfjr+pk/rαkjxdx1/pk.
This proves the final result.Corollary 3.
Under the assumption of Theorem7, taking s=m,αkj=−r/pkt for j≠k, and αkk=t1−r/pk with t∈ℝ, we have(47)∏j=1mfjxr≤∏k=1s∫Ω∏j=1mfjxr−tfkxt/r1/pk.Remarks:(i)
Takings=m,αkj=−r2/pk for j≠k, and αkk=r−r2/pk, inequality (47) is reduced to (43)(ii)
If we discuss Hölder’s inequality on the measure space with the counting measure, Hölder’s inequality for integrals is reduced to the corresponding result for sum(iii)
Instituting eachA∈A for Ω, the conclusion still holds since A,A∩A,μ is an induced measure spaceSimilar to equation (9), we derive the following theorem.Theorem 8.
Letr∈ℝ,fjx>0,rpk>0,αkj∈ℝ,∑k=1s1/pk=1/r, and ∑k=1s=0. Then,(48)∏j=1mfjxr≤ϕc≤∏k=1s∫Ω∏j=1mfjr+pk/rαkjxdx1/pk,where(49)ϕc=∫Ω∏j=1mfjrx1/r+∏k=1s∫Ω∏j=1mfjr+pk/rαkjxdx1/pk.


## 3.1. Generalized Hölder’s Inequality with a Pair ofr-Conjugate Exponents

In this part, we will generalize the celebrated Young inequality and Hölder inequality for integrals to those withr-conjugate exponents.Theorem 1.
(generalized Young inequality). Assume thatp and q are a pair of r-conjugate exponents. Then, for positive real numbers a and b,(19)abrr≤app+bqq,with equality if and only if ap=bq.Proof.
Puttingfx=ex, we have f′x=f″x=ex>0 for any real number x∈ℝ. Thus, fx is convex for x∈0,+∞. Assuming x=plna and y=qlnb, due to Jensen’s inequality, we obtain(20)frpx+rqy≤rpfx+rqfy.
Substituting, respectively,plna and qlnb for x and y, it follows that(21)erx/p+ry/q≤rpex+rqey,that is, abr/r≤ap/p+bq/q. The equality holds if and only if rx=ry or equally, ap=bq.
Then, we will derive the generalized Hölder inequality with a pair of conjugate exponents.Theorem 2.
(Hölder’s inequality with a pair of conjugate exponents). Letp and q be a pair of conjugate exponents. For any measurable functions f and g on some measure space Ω,A,μ, we have(22)fg1≤fpgq,with equality if and only if there exist two constants λ1 and λ2 which need not vanish such that λ1fp+λ2gq=0μ a.e.Proof.
One could refer to [22] when the need arises.Corollary 1.
For every measurable functionf on some measure space Ω,A,μ, we have(23)f1≤fp,for 1<p<∞.Proof.
It is trivial to prove this conclusion by puttingg=1 in Theorem 2.
The simplest measure on a measurable spaceΩ,A is the unit masses or Dirac measure δx,x∈Ω, defined by δxA=1Ax. We may form, for any countable set A=x1,x2,…, the associated counting measure μ=∑nδxn.
Every measurable mappingf of Ω into some measurable space S,S is referred to as a random element in S. A random element in S is called a random variable when S=ℝ, a random vector when S=ℝn, and a random sequence when S=ℝ∞, respectively.
Ifμ is Lebesgue measure on ℝn, we write Lpℝn instead of LpΩ,A,μ. If μ is the counting measure on a set A, it is customary to denote the corresponding Lp-space by ℒpA. In that case, the p-norm of a random vector x=x1,x2,…,xn or a random sequence x=x1,x2,… is xp=∑k=1nxkp1/p or xp=∑k=1∞xkp1/p, respectively. Thus, the Hölder inequality for sums is similar as that for integrals.Theorem 3.
(Hölder’s inequality with a pair ofr-conjugate exponents). Let p and q be a pair of r-conjugate exponents. For any measurable functions f and g on some measure space Ω,A,μ, we have(24)fgr≤fpgq,with equality if and only if there exist two constants λ1 and λ2 which need not vanish such that λ1fp+λ2gq=0μ a.e.Proof.
By Theorem2 with r/p+r/q=1, we have(25)fg1≤fp/rgq/r.
This impliesμfg≤μfp/rr/pμgq/rr/q, and furthermore, μfg1/r≤μfp/r1/pμgq/r1/q.
Replacingf and g by fr and fr, respectively, we obtain(26)fgr=μfgr1/r≤μfp1/pμgq1/q=fpgq.
Then, the equality holds in equation (18) if and only if λ1fp/r+λ2gq/r=0. Replacing f and g by fr and gr, respectively, we obtain λ1fp+λ2gq=0. Hence, the conclusion follows.Corollary 2.
For any measurable functionf on some measure space Ω,A,μ and nonnegative real numbers p and q with p≤q, we have(27)fp≤fq.Proof.
First, note that bothr<p and r<q hold when p and q are a pair of r-conjugate exponents. Then, putting g=1 in Theorem 5, we derive fr≤fp. Therefore, it follows that fp≤gq with p≤q.
Remark: Corollary2 implies that Lp-spaces are nonincreasing with respect to p.


## 3.2. Interpolation of Hölder’s Inequality withr-Conjugate Exponents

In literature [26], İşcan derived the interpolation of Hölder’s inequality with a pair of conjugate exponents with respect to integrable functions on u,v⊂ℝ. Furthermore, Gowda [27] gave an interpolation theorem relative to spectral norms. We will extend the interpolation of Hölder’s inequality to the r-conjugate exponents’ case on measure space Ω,A,μ.Definition 7.
A nondecreasing and countably subadditive set functionμ:2Ω⟶ℝ¯+ with μ∅=0 being referred to as an outer measure on a space Ω. Furthermore, given an outer measure μ on Ω, we say that a set A⊂Ω is μ-measurable if(28)μE=μE∩A+μE∩Ac,E⊂Ω.Theorem 4.
Assumep and q to be a pair of conjugate exponents and both f and g to be measurable functions on LpΩ,A,μ. Then, for any μ-measurable set A∈A, it follows that(29)fg1≤μAμΩ1/pfpμAμΩ1/qgq+μAcμΩ1/pfpμAcμΩ1/qgq≤fpgq.Proof.
Since setA is μ-measurable, it follows that μΩ=μΩ∩A+μΩ∩Ac=μA+μAc.
We first prove the left inequality of equation (23). By Theorem 2, we may obtain(30)fg1=1μΩ∫ΩμAfgdμ+∫ΩμAcfgdμ=1μΩ∫ΩμA1/pfμA1/qgdμ+∫ΩμAc1/pfμAc1/qgdμ≤1μΩ∫ΩμAfpdμ1/p∫ΩμAgqdμ1/q+∫ΩμAcfpdμ1/p∫ΩμAcgqdμ1/q=μAμΩ1/pfpμAμΩ1/qgq+μAcμΩ1/pfpμAcμΩ1/qgq.
Hence, the left inequality of equation (23) follows.
Furthermore, iffpgq=0, then f=0μ a.e. or g=0μ a.e. Thus, we have either fp=0 and ⋅fp=0 or gq=0 and ⋅gq=0. Therefore, the right inequality of equation (23) holds, that is,(31)μAμΩ1/pfpμAμΩ1/qgq+μAcμΩ1/pfpμAcμΩ1/qgq=fpgq.
Now, we consider the case ofM=fpgq≠0. By Young’s inequality, we derive(32)1μΩMμA1/pfpμA1/qgq+μAc1/pfpμAc1/qgq=1μΩ∫ΩμAfpdμ∫Ωfpdμ1/p∫ΩμAgqdμ∫Ωgqdμ1/q+∫ΩμAcfpdμ∫Ωfpdμ1/p∫ΩμAcgqdμ∫Ωgqdμ1/q≤1μΩ∫ΩμAfpdμp∫Ωfpdμ+∫ΩμAgqdμq∫Ωgqdμ+∫ΩμAcfpdμp∫Ωfpdμ+∫ΩμAcgqdμq∫Ωgqdμ=1p+1q=1.
Hence, the right inequality of equation (23) holds, that is,(33)1μΩμA1/pfpμA1/qgq+μAc1/pfpμAc1/qgq≤fpgq.
Therefore, equation (23) follows.Theorem 5.
Assumep and q to be a pair of r-conjugate exponents and both f and g to be measurable functions on LpΩ,A,μ. Then, for any μ-measurable set A∈A, it follows that(34)fgrr≤μAμΩ1/pfprμAμΩ1/qgqr+μAcμΩ1/pfprμAcμΩ1/qgqr≤fprgqr.Proof.
Letp and q be a pair of r-conjugate exponents, i.e., r/p+r/q=1. By Theorem 6, we derive(35)fg1≤μAμΩr/pfp/rμAμΩr/qgq/r+μAcμΩr/pfp/rμAcμΩr/qgq/r≤fp/rgq/r.
It follows that(36)∫Ωfgdμ≤∫ΩμAμΩfp/rdμr/p∫ΩμAμΩgq/rdμr/q+∫ΩμAcμΩfp/rdμr/p∫ΩμAcμΩgq/rdμr/q≤∫Ωfp/rdμr/p∫Ωgq/rdμr/q.
Replacingf and g by fr and gr, respectively, it follows that(37)∫Ωfgrdμ≤∫ΩμAμΩfpdμr/p∫ΩμAμΩgqdμr/q+∫ΩμAcμΩfpdμr/p∫ΩμAcμΩgqdμr/q≤∫Ωfpdμr/p∫Ωgqdμr/q,that is,(38)fgrr≤μAμΩ1/pfprμAμΩ1/qfqr+μAcμΩ1/pfprμAcμΩ1/qfqr≤fprgqr.
This completes the proof.


## 3.3. Generalized Hölder’s Inequality withs-Order r-Conjugate Exponents

In this part, we will present the most important results of this paper in the following theorems.Theorem 6.
(Hölder’s inequality withs-order r-conjugate exponents). Assume that pk,k=1,2,…,s, are s-order r-conjugate exponents and that fjx>0,j=1,2,…,m, are all measurable functions on LpΩ,A,μ. Then,(39)∏k=1sfxkr≤∏k=1sfkxpk.Proof.
We prove this conclusion by induction.
Ifs=2, the result reduces to Hölder’s inequality with a pair of conjugate exponents.
Assume that the conclusion holds whens=l. Then, putting s=l+1, we have(40)∏k=1l+1fkxr=∏k=1lfkxfl+1xr≤∏k=1lfkxqfl+1xpl+1,where 1/p1+1/p2+⋯+1/pl=1/q.
By the aforementioned assumption, it follows that(41)∏k=1lfkxq=∫Ω∏k=1lfkxdx1/q≤∏k=1l∫Ωfkqxpk/qdxq/pk1/q=∏k=1l∫Ωfkpkx1/pk.
Thus, combining the two inequalities above, we derive(42)∏k=1l+1fkxr≤∏k=1l∫Ωfkpkxdx1/pkfl+1xpl+1=∏k=1l+1fkxpk.
This proves the cases=l+1.Theorem 7.
(generalized Hölder’s inequality withs-order r-conjugate exponents). Assume that pk,k=1,2,…,s, are s-order r-conjugate exponents, αkj∈ℝ,j=1,2,…,m, and fjx>0,j=1,2,…,m, are all measurable functions on LpΩ,A,μ. Then,(43)∏j=1mfjxr≤∏k=1s∫Ω∏j=1mfjr+pk/rαkjxdx1/pk.Proof.
Assume the positive function sequence(44)gkx=∏j=1mfjr+pk/rαkjx1/pk,k=1,2,…,s.
By simple computation and rearrangement, it follows that(45)∫Ω∏k=1sgkrxdx=∫Ω∏k=1s∏j=1mfjr+pk/rαkjr/pkdx=∫Ω∏j=1mfjrxdx,with the assumptions ∑k=1s1/pk=1/r and ∑k=1sαkj=0.
Then, by using Hölder’s inequality withs-order r-conjugate exponents, it follows that(46)∏j=1mfjxr=∏k=1sgkxr≤∏k=1sgkxpk=∏k=1s∫Ω∏j=1mfjr+pk/rαkjxdx1/pk.
This proves the final result.Corollary 3.
Under the assumption of Theorem7, taking s=m,αkj=−r/pkt for j≠k, and αkk=t1−r/pk with t∈ℝ, we have(47)∏j=1mfjxr≤∏k=1s∫Ω∏j=1mfjxr−tfkxt/r1/pk.Remarks:(i)
Takings=m,αkj=−r2/pk for j≠k, and αkk=r−r2/pk, inequality (47) is reduced to (43)(ii)
If we discuss Hölder’s inequality on the measure space with the counting measure, Hölder’s inequality for integrals is reduced to the corresponding result for sum(iii)
Instituting eachA∈A for Ω, the conclusion still holds since A,A∩A,μ is an induced measure spaceSimilar to equation (9), we derive the following theorem.Theorem 8.
Letr∈ℝ,fjx>0,rpk>0,αkj∈ℝ,∑k=1s1/pk=1/r, and ∑k=1s=0. Then,(48)∏j=1mfjxr≤ϕc≤∏k=1s∫Ω∏j=1mfjr+pk/rαkjxdx1/pk,where(49)ϕc=∫Ω∏j=1mfjrx1/r+∏k=1s∫Ω∏j=1mfjr+pk/rαkjxdx1/pk.


## 4. Conclusion and Future Work

The classical Hölder’s inequality of the form(50)∫ab∏j=1mfjxdx≤∏j=1m∫abfjpjxdx1/pjis applied widely in social science, management science, and natural science, where ∑j=1m1/pj=1. Qiang and Hu [20] expanded this result to the case ∑k=1m1/pk=r and derived(51)∫ab∏j=1mfjxdx≤ϕc≤∏k=1s∫ab∏j=1mfj1+rpkαkjxdx1/rpk,where(52)ϕc=∫ac∏j=1mfjx+∏k=1s∫cb∏j=1mfj1+rpkαkjxdx1/rpkis a nonincreasing function with a≤c≤b.On the one hand, in modern probability theory, it is customary to relate all objects of the study to a fundamental probability spaceΩ,A,P; especially, Lp-space is a most important probability space. On the other hand, to seek the relationship between inequality (52) and the mean value theorem of integrals is interesting. Thus, we want to investigate generalized Hölder’s inequality with r-conjugate exponents under the Lp-space. Successively, based on r-conjugate exponents, we derived completely generalized Young’s inequality, generalized Hölder’s inequality, the interpolation of Hölder’s inequality, and generalized s-order Hölder’s inequality.Regretfully, the relationship between the mean value theorem of integrals and the interpolation of Hölder’s inequality is left to consider intensively as the future work.



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