MPEMathematical Problems in Engineering1563-51471024-123XHindawi10.1155/2021/55344695534469Research ArticleThe Improvement of Hölder’s Inequality with r-Conjugate Exponents Relative to the Lp-Normhttps://orcid.org/0000-0002-5878-5897LiangXiaojunWanAyingHattafKhalidSchool of Mathematics & StatisticsHulunbuir CollegeHulunbeier 021000Inner MongoliaChina2021304202120212612021542021164202130420212021Copyright © 2021 Xiaojun Liang and Aying Wan.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.

This paper investigates Hölder’s inequality under the condition of r-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.

Hulunbuir College Doctor Foundation Project of China2018BS162020BS04Natural Science Foundation of Inner Mongolia2018MS01023National Natural Science Foundation of China12061033
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 functions f and g on some measurable space Ω,A,μ, it follows that(1)fg1fpgq,for all p,q>0 with 1/p+1/q=1, where fp=μfp1/p=fpdμ<, for all measurable functions f:Ω (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 exist c, ε>0 only depending on n, p, and Λ so that(2)Qr,rpup+εdxdtcQ2r,2rpup2rp+updxdtεQ2r,2rpupdxdt,for all time-space cylinders with Q2r,2rp=I2rp×B2r0,T×Ω, where u0. The recent representative articles with respect to this topic are referred to in .

The other refinements are Hölder’s inequality and inverse Hölder’s inequality for the pseudo-integral established by Agahi et al. . Assume that p and q are a pair of conjugate exponents and that Ω,A,μ is a measure space. Suppose that f,g:Ωa,b are two measurable functions and that a generator g:a,b0, of the pseudo-addition and the pseudo-multiplication is an increasing function. Then, for any σ--measure μ, it follows that(3)ΩfgdμΩfpdμ1/pΩgqdμ1/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 ) since fuzzy measures and Sugeno integral have been successfully applied to various fields such as to decision-making  and to artificial intelligence . Besides, Khan et al. investigated the converses of the Jensen inequality in their articles such as .

The fundamental refinement is the improvement of Hölder’s inequality itself. As far as in 1961, Beckenbach and Bellman  derived generalized Hölder’s inequality of the following form: let m and n be positive integers, and let aij>0,1in,1jm, pj>0, and i=1m1/pj=1. Then, it follows that(4)i=1nj=1maijj=1mi=1naijpj1/pj.

The corresponding integral form is(5)abj=1mfjxdxj=1mabfjpjxdx1/pj,where fjx>0,j=1,2,,m, xa,b, <a<b<,pj>0,j=1m1/pj=1, and fjLpja,b. Note that, by taking m=2, f1x=fx, and f2x=gx, inequality (5) reduces to inequality (1). Qiang and Hu , then, derived further contributions on this topic as follows: let aij>0, pk>0, αij, i=1,2,,n;j=1,2,,m;k=1,2,,s, k=1s1/pk=1, and k=1sαij=0. Then,(6)i=1nj=1maijk=1si=1nj=1maij1+pkαkj1/pk.

Moreover, for the integral form of the above inequality, if fjx>0,j=1,2,,m,xa,b,<a<b<, and fjCa,b, then(7)abj=1mfjxdxk=1sabj=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 jk, and αkk=11/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  established an extension of the Callebaut inequality , that is,(8)abj=1mfjxdxabj=1mfj1+αjxdx1/2abj=1mfj1αjxdx1/2,where αj and fjx>0, i=1,2,,n;j=1,2,,m.

Based on these contributions of Masjed-Jamei, Qiang and Hu  derived the following interesting result:(9)abj=1mfjxdxϕck=1sabj=1mfj1+rpkαkjxdx1/rpk,where(10)ϕc=acj=1mfjxdx+k=1scbj=1mfj1+rpkαkjxdx1/rpkis a nonincreasing function with acb.

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 ξa,b such that(11)abfxgxdx=gaaξfxdx+gbξbfxdx.

Putting m=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,at1t2b.

In this paper, our contributions are concluded as follows. We have derived generalized Hölder’s inequality under the condition of r-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 with s-order r-conjugate exponents.

Definition 1.

If p 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 p1. Hence, it is reasonable to regard 1 and as a pair of conjugate exponents .

Definition 2.

If p1,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.

If p 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.

If p1,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 the p-norm.

Definition 5.

The integral of a real-valued, measurable function f on some measure space Ω,A,μ is defined as(16)μf=fdμ=fωμdω.

If f is simple and nonnegative, hence of the form c11A1++cn1An for some nZ+, A1,,AnA, and c1,,cn+, μf is defined as(17)μf=c1μA1+c2μA2++cnμAn.

For any nonnegative measurable function f, we may choose some simple measurable functions f1,f2, with 0fnf, and define μf=limnμfn .

Note that it is customary to relate all objects of the study to a basic probability space Ω,A,μ, which is nothing more than a normalized measure space.

Definition 6.

Given a measure space Ω,A,μ and p1, 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 Results3.1. Generalized Hölder’s Inequality with a Pair of <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M143"><mml:mi>r</mml:mi></mml:math></inline-formula>-Conjugate Exponents

In this part, we will generalize the celebrated Young inequality and Hölder inequality for integrals to those with r-conjugate exponents.

Theorem 1.

(generalized Young inequality). Assume that p and q are a pair of r-conjugate exponents. Then, for positive real numbers a and b,(19)abrrapp+bqq,with equality if and only if ap=bq.

Proof.

Putting fx=ex, we have fx=fx=ex>0 for any real number x. Thus, fx is convex for x0,+. Assuming x=plna and y=qlnb, due to Jensen’s inequality, we obtain(20)frpx+rqyrpfx+rqfy.

Substituting, respectively, plna and qlnb for x and y, it follows that(21)erx/p+ry/qrpex+rqey,that is, abr/rap/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). Let p 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)fg1fpgq,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  when the need arises.

Corollary 1.

For every measurable function f on some measure space Ω,A,μ, we have(23)f1fp,for 1<p<.

Proof.

It is trivial to prove this conclusion by putting g=1 in Theorem 2.

The simplest measure on a measurable space Ω,A is the unit masses or Dirac measure δx,xΩ, defined by δxA=1Ax. We may form, for any countable set A=x1,x2,, the associated counting measure μ=nδxn.

Every measurable mapping f 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=, a random vector when S=n, and a random sequence when S=, respectively.

If μ is Lebesgue measure on n, we write Lpn 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 pA. 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=1xkp1/p, respectively. Thus, the Hölder inequality for sums is similar as that for integrals.

Theorem 3.

(Hölder’s inequality with a pair of r-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)fgrfpgq,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 Theorem 2 with r/p+r/q=1, we have(25)fg1fp/rgq/r.

This implies μfgμfp/rr/pμgq/rr/q, and furthermore, μfg1/rμfp/r1/pμgq/r1/q.

Replacing f and g by fr and fr, respectively, we obtain(26)fgr=μfgr1/rμfp1/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 function f on some measure space Ω,A,μ and nonnegative real numbers p and q with pq, we have(27)fpfq.

Proof.

First, note that both r<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 frfp. Therefore, it follows that fpgq with pq.

Remark: Corollary 2 implies that Lp-spaces are nonincreasing with respect to p.

3.2. Interpolation of Hölder’s Inequality with <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M251"><mml:mi>r</mml:mi></mml:math></inline-formula>-Conjugate Exponents

In literature , İşcan derived the interpolation of Hölder’s inequality with a pair of conjugate exponents with respect to integrable functions on u,v. Furthermore, Gowda  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 function μ:2Ω¯+ 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 AΩ is μ-measurable if(28)μE=μEA+μEAc,EΩ.

Theorem 4.

Assume p 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 AA, it follows that(29)fg1μAμΩ1/pfpμAμΩ1/qgq+μAcμΩ1/pfpμAcμΩ1/qgqfpgq.

Proof.

Since set A is μ-measurable, it follows that μΩ=μΩA+μΩAc=μA+μAc.

We first prove the left inequality of equation (23). By Theorem 2, we may obtain(30)fg1=1μΩΩμAfgdμ+ΩμAcfgdμ=1μΩΩμA1/pfμA1/qgdμ+ΩμAc1/pfμAc1/qgdμ1μΩΩμAfpdμ1/pΩμAgqdμ1/q+ΩμAcfpdμ1/pΩμAcgqdμ1/q=μAμΩ1/pfpμAμΩ1/qgq+μAcμΩ1/pfpμAcμΩ1/qgq.

Hence, the left inequality of equation (23) follows.

Furthermore, if fpgq=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 of M=fpgq0. By Young’s inequality, we derive(32)1μΩMμA1/pfpμA1/qgq+μAc1/pfpμAc1/qgq=1μΩΩμAfpdμΩfpdμ1/pΩμAgqdμΩgqdμ1/q+ΩμAcfpdμΩfpdμ1/pΩμAcgqdμΩgqdμ1/q1μΩΩμAfpdμpΩfpdμ+ΩμAgqdμqΩgqdμ+ΩμAcfpdμpΩfpdμ+ΩμAcgqdμqΩgqdμ=1p+1q=1.

Hence, the right inequality of equation (23) holds, that is,(33)1μΩμA1/pfpμA1/qgq+μAc1/pfpμAc1/qgqfpgq.

Therefore, equation (23) follows.

Theorem 5.

Assume p 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 AA, it follows that(34)fgrrμAμΩ1/pfprμAμΩ1/qgqr+μAcμΩ1/pfprμAcμΩ1/qgqrfprgqr.

Proof.

Let p and q be a pair of r-conjugate exponents, i.e., r/p+r/q=1. By Theorem 6, we derive(35)fg1μAμΩr/pfp/rμAμΩr/qgq/r+μAcμΩr/pfp/rμAcμΩr/qgq/rfp/rgq/r.

It follows that(36)ΩfgdμΩμAμΩfp/rdμr/pΩμAμΩgq/rdμr/q+ΩμAcμΩfp/rdμr/pΩμAcμΩgq/rdμr/qΩfp/rdμr/pΩgq/rdμr/q.

Replacing f and g by fr and gr, respectively, it follows that(37)ΩfgrdμΩμAμΩfpdμr/pΩμAμΩgqdμr/q+ΩμAcμΩfpdμr/pΩμAcμΩgqdμr/qΩfpdμr/pΩgqdμr/q,that is,(38)fgrrμAμΩ1/pfprμAμΩ1/qfqr+μAcμΩ1/pfprμAcμΩ1/qfqrfprgqr.

This completes the proof.

3.3. Generalized Hölder’s Inequality with <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M307"><mml:mi>s</mml:mi></mml:math></inline-formula>-Order <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M308"><mml:mi>r</mml:mi></mml:math></inline-formula>-Conjugate Exponents

In this part, we will present the most important results of this paper in the following theorems.

Theorem 6.

(Hölder’s inequality with s-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=1sfxkrk=1sfkxpk.

Proof.

We prove this conclusion by induction.

If s=2, the result reduces to Hölder’s inequality with a pair of conjugate exponents.

Assume that the conclusion holds when s=l. Then, putting s=l+1, we have(40)k=1l+1fkxr=k=1lfkxfl+1xrk=1lfkxqfl+1xpl+1,where 1/p1+1/p2++1/pl=1/q.

By the aforementioned assumption, it follows that(41)k=1lfkxq=Ωk=1lfkxdx1/qk=1lΩfkqxpk/qdxq/pk1/q=k=1lΩfkpkx1/pk.

Thus, combining the two inequalities above, we derive(42)k=1l+1fkxrk=1lΩfkpkxdx1/pkfl+1xpl+1=k=1l+1fkxpk.

This proves the case s=l+1.

Theorem 7.

(generalized Hölder’s inequality with s-order r-conjugate exponents). Assume that pk,k=1,2,,s, are s-order r-conjugate exponents, αkj,j=1,2,,m, and fjx>0,j=1,2,,m, are all measurable functions on LpΩ,A,μ. Then,(43)j=1mfjxrk=1sΩj=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)Ωk=1sgkrxdx=Ωk=1sj=1mfjr+pk/rαkjr/pkdx=Ωj=1mfjrxdx,with the assumptions k=1s1/pk=1/r and k=1sαkj=0.

Then, by using Hölder’s inequality with s-order r-conjugate exponents, it follows that(46)j=1mfjxr=k=1sgkxrk=1sgkxpk=k=1sΩj=1mfjr+pk/rαkjxdx1/pk.

This proves the final result.

Corollary 3.

Under the assumption of Theorem 7, taking s=m,αkj=r/pkt for jk, and αkk=t1r/pk with t, we have(47)j=1mfjxrk=1sΩj=1mfjxrtfkxt/r1/pk.

Remarks:

Taking s=m,αkj=r2/pk for jk, and αkk=rr2/pk, inequality (47) is reduced to (43)

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

Instituting each AA for Ω, the conclusion still holds since A,AA,μ is an induced measure space

Similar to equation (9), we derive the following theorem.

Theorem 8.

Let r,fjx>0,rpk>0,αkj,k=1s1/pk=1/r, and k=1s=0. Then,(48)j=1mfjxrϕck=1sΩj=1mfjr+pk/rαkjxdx1/pk,where(49)ϕc=Ωj=1mfjrx1/r+k=1sΩj=1mfjr+pk/rαkjxdx1/pk.

4. Conclusion and Future Work

The classical Hölder’s inequality of the form(50)abj=1mfjxdxj=1mabfjpjxdx1/pjis applied widely in social science, management science, and natural science, where j=1m1/pj=1. Qiang and Hu  expanded this result to the case k=1m1/pk=r and derived(51)abj=1mfjxdxϕck=1sabj=1mfj1+rpkαkjxdx1/rpk,where(52)ϕc=acj=1mfjx+k=1scbj=1mfj1+rpkαkjxdx1/rpkis a nonincreasing function with acb.

On the one hand, in modern probability theory, it is customary to relate all objects of the study to a fundamental probability space Ω,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.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This paper was supported by Hulunbuir College Doctor Foundation Project of China (2018BS16 and 2020BS04), the Natural Science Foundation of Inner Mongolia (CN) (2018MS01023), and the National Natural Science Foundation of China (12061033).

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