# On a New Extension of Mulholland’s Inequality in the Whole Plane

Authors: Bicheng Yang; Yanru Zhong; Qiang Chen
Journal: Journal of Function Spaces
 (2018)

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## Abstract

A new, more accurate extension of Mulholland’s inequality in the whole plane with a best possible constant factor is presented by introducing independent parameters, applying weight coefficients and using Hermite-Hadamard’s inequality. Moreover, the equivalent forms, some particular cases, and the operator expressions are considered.


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## Body

## 1. Introduction

Assume thatp>1, 1/p+1/q=1, am,bn≥0, 0<∑m=1∞amp<∞, and 0<∑n=1∞bnq<∞; then Hardy-Hilbert’s inequality (cf. [1]) is as follows:(1)∑n=1∞∑m=1∞ambnm+n<πsin⁡π/p∑m=1∞amp1/p∑n=1∞bnq1/q,where π/sin⁡(π/p) is the best possible constant factor. Replacing am/m and bn/n byam andbn, respectively, yields the following Mulholland’s inequality (cf. Theorem 343 in [1]):(2)∑n=2∞∑m=2∞ambnln⁡mn<πsin⁡π/p∑m=2∞ampm1/p∑n=2∞bnqn1/q.(1) and (2) are two important inequalities in analysis and its applications (cf. [1, 2]).In 2007, Yang [3] firstly provided the following Hilbert-type integral inequality in the whole plane:(3)∫-∞∞∫-∞∞fxgy1+ex+yλdxdy<Bλ2,λ2∫-∞∞e-λxf2xdx∫-∞∞e-λyg2ydy1/2,where Bλ/2,λ/2λ>0 is the best possible constant factor. Various extensions of inequalities (1)–(3) (cf. [4–15]) have been presented since then. Recently, Yang and Chen [16] presented an extension of (1) in the whole plane as follows:(4)∑n=1∞∑m=1∞ambnm-ξ+n-ηλ<2Bλ1,λ2∑m=1∞m-ξp1-λ1-1amp1/p∑n=1∞n-ηq1-λ2-1bnq1/q,where 2Bλ1,λ20<λ1,λ2≤1,λ1+λ2=λ,ξ,η∈0,1/2 is the best possible constant factor. In addition to Yang and Chen, Xin et al. [17] have also carried out a similar work.In this paper, we present a new, more accurate extension of (2) in the whole plane with a best possible constant factor that is similar to that in (4) via introducing independent parameters, applying weight coefficients, and using Hermite-Hadamard’s inequality. Moreover, the equivalent forms, some particular cases, and the operator expressions are considered.


## 2. Two Lemmas

In this section, we assume thatp>1, 1/p+1/q=1, λ1,λ2>0, λ1+λ2=λ≤1, ξ,η∈0,1/2, α,β∈[arccos⁡1/3,π/2],and(5)hνλ1≔2πcsc2⁡γλsin⁡πλ1/λγ=α,β.Remark 1.
In view of the conditions thatα,β∈[arccos⁡1/3,π/2],ξ,η∈0,1/2, it follows that(6)32±η1∓cos⁡β≥1,32±ξ1∓cos⁡α≥1.
Forx,y≥3/2, let the function(7)hx,y≔1lnλ⁡x-ξ+x-ξcos⁡α+lnλ⁡y-η+y-ηcos⁡β.We define two weight coefficients as follows:(8)ωλ2,m≔∑n=2∞hm,nn-η+n-ηcos⁡β·lnλ1⁡m-ξ+m-ξcos⁡αln1-λ2⁡n-η+n-ηcos⁡β,m∈N∖1,(9)ϖλ1,n≔∑m=2∞hm,nm-ξ+m-ξcos⁡α·lnλ2⁡n-η+n-ηcos⁡βln1-λ1⁡m-ξ+m-ξcos⁡α,n∈N∖1,where ∑j=2∞⋯=∑j=-2-∞⋯+∑j=2+∞⋯j=m,n.Lemma 2.
The inequalities(10)hβλ11-θλ2,m<ωλ2,m<hβλ1,m∈N∖1,are valid, where(11)θλ2,m≔λπsin⁡πλ1λ∫0ln⁡2+η1+cosβ/ln⁡m-ξ+m-ξcos⁡αuλ2-11+uλdu=O1lnλ2⁡m-ξ+m-ξcos⁡α∈0,1.Proof.
Form∈N∖1, let Aξm≔m-ξ+m-ξcos⁡α and(12)h1m,y≔1lnλ⁡Aξm+lnλ⁡y-ηcos⁡β-1,y<-32,h2m,y≔1lnλ⁡Aξm+lnλ⁡y-ηcos⁡β+1,y>32.Then(13)h1m,-y=1lnλ⁡Aξm+lnλ⁡y+η1-cos⁡β,y>32,yields(14)ωλ2,m=∑n=-2-∞h1m,nlnλ1Aξmn-ηcosβ-1ln1-λ2n-ηcosβ-1+∑n=2∞h2m,nlnλ1⁡Aξmn-η1+cos⁡βln1-λ2⁡n-η1+cos⁡β=lnλ1⁡Aξm1-cos⁡β∑n=2∞h1m,-nn+ηln1-λ2n+η1-cosβ+lnλ1Aξm1+cos⁡β∑n=2∞h2m,nn-ηln1-λ2⁡n-η1+cos⁡β.In virtue of 0<λ≤1, 0<λ2<1, we find that, for y>3/2,(15)ddyhim,-1iyy--1iηln1-λ2⁡y--1iη1+-1icos⁡β<0,d2dy2him,-1iyy--1iηln1-λ2⁡y--1iη1+-1icos⁡β>0i=1,2,and it follows that (16)him,-1iyy--1iηln1-λ2⁡y--1iη1+-1icos⁡βi=1,2are strict decreasing and strict convex in (3/2,∞). Then Hermite-Hadamard’s inequality (cf. [18]) and (14) yield(17)ωλ2,m<lnλ1⁡Aξm1-cos⁡β∫3/2∞h1m,-yy+ηln1-λ2⁡y+η1-cos⁡βdy+lnλ1⁡Aξm1+cos⁡β∫3/2∞h2m,yy-ηln1-λ2⁡y-η1+cos⁡βdy.Setting u=ln⁡[(y+η)(1-cosβ)]/ln⁡Aξm(u=ln⁡[(y-η)(1+cosβ)]/ln⁡Aξm) in the above first (second) integral, in view of Remark 1 we obtain(18)ωλ2,m<11-cos⁡β+11+cos⁡β∫0∞uλ2-1du1+uλ=2csc2⁡βλ∫0∞vλ2/λ-1dv1+v=2πcsc2⁡βλsin⁡πλ1/λ=hβλ1,by simplifications. Similarly, (14) yields(19)ωλ2,m>lnλ1⁡Aξm1-cos⁡β∫2∞h1m,-yy+ηln1-λ2⁡y+η1-cos⁡βdy+lnλ1⁡Aξm1+cos⁡β∫2∞h2m,yy-ηln1-λ2⁡y-η1+cos⁡βdy≥11-cos⁡β+11+cos⁡β∫ln⁡2+η1+cos⁡β/ln⁡Aξm∞uλ2-1du1+uλ=hβλ1-2csc2⁡β∫0ln⁡2+η1+cos⁡β/ln⁡Aξmuλ2-1du1+uλ=hβλ11-θλ2,m>0,according to monotonicity, where θλ2,m is indicated by (11). It follows that θλ2,m<1 and(20)0<θλ2,m<λπsin⁡πλ1λ∫0ln⁡2+η1+cos⁡β/ln⁡m-ξ+m-ξcos⁡αuλ2-1du=λπsin⁡πλ1λln⁡2+η1+cos⁡βln⁡m-ξ+m-ξcos⁡αλ2.Hence, (10) and (11) are valid.Similarly, we have the following.Lemma 3.
For0<λ≤1, 0<λ1<1, the inequalities(21)hαλ11-θ~λ1,n<ϖλ1,n<hαλ1,n∈N∖1,are valid, where(22)θ~λ1,n≔λπsin⁡πλ1λ∫0ln⁡2+ξ1+cos⁡α/ln⁡n-η+n-ηcos⁡βuλ1-11+uλdu=O1lnλ1⁡n-η+n-ηcos⁡β∈0,1.Lemma 4.
Ifς,γ=ξ,αorη,β,ρ>0, then we have(23)Hρς,γ≔∑n=2∞ln-1-ρ⁡n-ς+n-ςcos⁡γn-ς+n-ςcos⁡γ=1ρ2csc2⁡γ+o1ρ⟶0+.Proof.
According to Hermite-Hadamard’s inequality, we obtain(24)Hρς,γ=∑n=-2-∞ln-1-ρ⁡n-ςcos⁡γ-1n-ςcos⁡γ-1+∑n=2∞ln-1-ρ⁡n-ςcos⁡γ+1n-ςcos⁡γ+1=∑n=2∞ln-1-ρ⁡n+ς1-cos⁡γn-ς1-cos⁡γ+ln-1-ρ⁡n-ςcos⁡γ+1n-ςcos⁡γ+1≤∫3/2∞ln-1-ρ⁡y+ς1-cos⁡γy-ς1-cos⁡γ+ln-1-ρ⁡y-ςcos⁡γ+1y-ςcos⁡γ+1dy=1ρln-ρ⁡3/2+ς1-cos⁡γ1-cos⁡γ+ln-ρ⁡3/2-ς1+cos⁡γ1+cos⁡γ=1ρ11-cos⁡γ+11+cos⁡γ+o11ρ⟶0+.This leads to(25)Hρς,γ=∑n=2∞ln-1-ρ⁡n+ς1-cos⁡γn-ς1-cos⁡γ+ln-1-ρ⁡n-ςcos⁡γ+1n-ςcos⁡γ+1≥∫2∞ln-1-ρ⁡y+ς1-cos⁡γy-ς1-cos⁡γ+ln-1-ρ⁡y-ςcos⁡γ+1y-ςcos⁡γ+1dy=1ρln-ρ⁡2+ς1-cos⁡γ1-cos⁡γ+ln-ρ⁡2-ς1+cos⁡γ1+cos⁡γ=1ρ11-cos⁡γ+11+cos⁡γ+o21ρ⟶0+.Therefore, (23) is valid.


## 3. Main Results

In this section, we also defineBηn≔n-η+n-ηcos⁡β, and(26)kλ1≔hβ1/pλ1hα1/qλ1=2πcsc2/p⁡βcsc2/qαλsin⁡πλ1/λ.Theorem 5.
Suppose thatam,bn≥0m,n∈N∖1,(27)0<∑m=2∞lnp1-λ1-1⁡AξmAξm1-pamp<∞,0<∑n=2∞lnq1-λ2-1⁡BηnBηn1-qbnq<∞.Then we obtain the following equivalent inequalities:(28)I≔∑n=2∞∑m=2∞hm,nambn<kλ1∑m=2∞lnp1-λ1-1AξmAξm1-pamp1/p∑n=2∞lnq1-λ2-1BηnBηn1-qbnq1/q,(29)J≔∑n=2∞lnpλ2-1BηnBηn∑m=2∞hm,namp1/p<kλ1∑m=2∞lnp1-λ1-1AξmAξm1-pamp1/p.Particularly, (i) for α=β=π/2, ξ,η∈0,1/2, we have(30)∑n=2∞∑m=2∞ambnlnλm-ξ+lnλn-η<2πλsin⁡πλ1/λ∑m=2∞lnp1-λ1-1m-ξm-ξ1-pamp1/p∑n=2∞lnq1-λ2-1n-ηn-η1-qbnq1/q,(31)∑n=2∞lnpλ2-1n-ηn-η∑m=2∞amlnλm-ξ+lnλn-ηp1/p<2πλsin⁡πλ1/λ∑m=2∞lnp1-λ1-1m-ξm-ξ1-pamp1/p.
(ii) Forξ=η=0, α,β∈arccos⁡1/3,π/2, we have(32)∑n=2∞∑m=2∞ambnlnλm+mcos⁡α+lnλn+ncos⁡β<kλ1∑m=2∞lnp1-λ1-1m+mcos⁡αm+mcos⁡α1-pamp1/p∑n=2∞lnq1-λ2-1n+ncos⁡βn+ncos⁡β1-qbnq1/q,∑n=2∞lnpλ2-1n+ncos⁡βn+ncos⁡β∑m=2∞amlnλm+mcos⁡α+lnλn+ncos⁡βp1/p<kλ1∑m=2∞lnp1-λ1-1m+mcos⁡αm+mcos⁡α1-pamp1/p.Proof.
According to Hölder’s inequality with weight (cf. [18]) and (9), we find that(33)∑m=2∞hm,namp=∑m=2∞hm,nAξm1/qln1-λ1/qAξmln1-λ2/pBηnamln1-λ2/pBηnAξm1/qln1-λ1/qAξmp≤∑m=2∞hm,nAξmp/qln1-λ1p/qAξmln1-λ2Bηnamp∑m=2∞hm,nln1-λ2q/pBηnAξmln1-λ1Aξmp-1=ϖλ1,np-1Bηnlnpλ2-1Bηn∑m=2∞hm,nAξmp/qln1-λ1p/qAξmBηnln1-λ2Bηnamp.Then (21) yields(34)J<hα1/qλ1∑n=2∞∑m=2∞hm,nAξmp/qln1-λ1p/qAξmBηnln1-λ2Bηnamp1/p=hα1/qλ1∑m=2∞∑n=2∞hm,nAξmp/qln1-λ1p/qAξmBηnln1-λ2Bηnamp1/p=hα1/qλ1∑m=2∞ωλ2,mnp1-λ1-1AξmAξm1-pamp1/p.Combining (10) and (26), we obtain (29).
Using Hölder’s inequality again, we obtain(35)I=∑n=2∞Bηn-1/pln1/p-λ2Bηn∑m=2∞hm,namln1/p-λ2BηnBηn-1/pbn≤J∑n=2∞lnq1-λ2-1BηnBηn1-qbnq1/q.Then according to (29), we obtain (28).
Further, assume that (28) is valid; let(36)bn≔lnpλ2-1BηnBηn∑m=2∞hm,namp-1,n∈N∖1,and find(37)J=∑n=2∞lnq1-λ2-1BηnBηn1-qbnq1/p.According to (34), it follows that J<∞. If J=0, then (30) is trivially valid; if J>0, then we have(38)0<∑n=2∞lnq1-λ2-1BηnBηn1-qbnq=Jp=I<kλ1∑m=2∞lnp1-λ1-1AξmAξm1-pamp1/p∑n=2∞lnq1-λ2-1BηnBηn1-qbnq1/q,J=∑n=2∞lnq1-λ2-1BηnBηn1-qbnq1/p<kλ1∑m=2∞lnp1-λ1-1AξmAξm1-pamp1/p.Thus, (29) is valid, which is equivalent to (28).Theorem 6.
With regard to the assumptions in Theorem5, kλ1 is the best possible constant factor in (28) and (29).Proof.
For0<ε<min⁡{q(1-λ1),qλ2}, let λ~1=λ1+ε/q, λ~2=λ2-ε/q∈0,1, and(39)a~m≔lnλ1-ε/p-1AξmAξm=lnλ~1-ε-1AξmAξmm∈N∖1,b~n≔lnλ2-ε/q-1BηnBηn=lnλ~2-1BηnBηnn∈N∖1.Then (23) and (21) yield(40)I~1≔∑m=2∞lnp1-λ1-1AξmAξm1-pa~mp1/p∑n=2∞lnq1-λ2-1BηnBηn1-qb~nq1/q=∑m=2∞ln-1-εAξmAξm1/p∑n=2∞ln-1-εBηnBηn1/q=1ε2csc2α+o11/p2csc2β+o~11/qε⟶0+,I~≔∑n=2∞∑m=2∞hm,na~mb~n=∑m=2∞∑n=2∞hm,nlnλ~1-ε-1AξmAξmlnλ~2-1BηnBηn=∑m=2∞ωλ~2,mln-ε-1AξmAξm≥hβλ~1∑m=2∞1-θλ~2,mln-ε-1AξmAξm=hβλ~1∑m=2∞ln-ε-1AξmAξm-∑m=2∞Oln-ε/p+λ2-1AξmAξm=1εhβλ~12csc2α+o1-εO1.
If there exists a positive numberK≤k(λ1), such that (28) is still valid when replacing k(λ1) by K, then we obtain(41)εI~=ε∑n=2∞∑m=2∞hm,na~mb~n<εKI~1.Hence, in view of the above results, it follows that(42)hβλ1+εq2csc2⁡α+o1-εO1<K2csc2⁡α+o11/p2csc2⁡β+o~11/q,and then(43)4πλsin⁡πλ1/λcsc2⁡βcsc2⁡α≤2Kcsc2/p⁡αcsc2/q⁡βε⟶0+;namely,(44)kλ1=2πλsin⁡πλ1/λcsc2/p⁡βcsc2/q⁡α≤K.Hence, K=kλ1 is the best possible constant factor in (28).
k
λ
1 in (29) is the best possible constant factor. Otherwise, we would obtain a contradiction according to (35) that kλ1 in (28) is not the best possible constant factor.


## 4. Operator Expressions

Letφm≔lnp1-λ1-1Aξm/Aξm1-p(m∈N∖{1}), and ψ(n):=lnq1-λ2-1Bηn/(Bη(n))1-q, where(45)ψ1-pn≔lnpλ2-1BηnBηnn∈N∖1.We define the real weighted normed function spaces as follows:(46)lp,φ≔a=amm=2∞;ap,φ=∑m=2∞φmamp1/p<∞,lq,ψ≔b=bnn=2∞;bq,ψ=∑n=2∞ψnbnq1/q<∞,lp,ψ1-p≔c=cnn=2∞;cp,ψ1-p=∑n=2∞ψ1-pncnp1/p<∞.For a=amm=2∞∈lp,φ, let cn=∑m=2∞hm,nam and c=cnn=2∞, and it follows from (29) that cp,ψ1-p<kλ1ap,φ; namely, c∈lp,ψ1-p.Further, we define a Mulholland-type operatorT:lp,φ→lp,ψ1-p as follows: for am≥0, a=amm=2∞∈lp,φ, there exists a unique representation Ta=c∈lp,ψ1-p. We also define the following formal inner product of Ta and b=bnn=2∞∈lq,ψbn≥0:(47)Ta,b≔∑n=2∞∑m=2∞hm,nambn.Hence, we can, respectively, rewrite (28) and (29) as the following operator expressions:(48)Ta,b<kλ1ap,φbq,ψ,Tap,ψ1-p<kλ1ap,φ.It follows that the operator T is bounded with(49)T≔supa≠θ∈lp,φTap,ψ1-pap,φ≤kλ1.Since kλ1 in (29) is the best possible constant factor, we obtain(50)T=kλ1=2πcsc2/p⁡βcsc2/q⁡αλsin⁡πλ1/λ.Remark 7.
(i) Forξ=η=0 in (30), we have the following new inequality:(51)∑n=2∞∑m=2∞ambnlnλm+lnλn<2πλsin⁡πλ1/λ∑m=2∞lnp1-λ1-1mm1-pamp1/p∑n=2∞lnq1-λ2-1nn1-qbnq1/q.It follows that (30) is a more accurate extension of (51).
(ii) Ifa-m=am,b-n=bnm,n∈N∖1, then (30) reduces to(52)∑n=2∞∑m=2∞1lnλm-ξ+lnλn-η+1lnλm-ξ+lnλn+η+1lnλm+ξ+lnλn-η+1lnλm+ξ+lnλn+ηambn<2πλsin⁡πλ1/λ∑m=2∞lnp1-λ1-1m-ξm-ξ1-p+lnp1-λ1-1m+ξm+ξ1-pamp1/p×∑n=2∞lnq1-λ2-1n-ηn-η1-q+lnq1-λ2-1n+ηn+η1-qbnq1/q.
(iii) Ifλ=1, λ1=1/q, λ2=1/p, ξ=η∈0,1/2, then (52) reduces to(53)∑n=2∞∑m=2∞1ln⁡m-ξn-ξ+1ln⁡m-ξn+ξ+1ln⁡m+ξn-ξ+1ln⁡m+ξn+ξambn<2πsin⁡π/p∑m=2∞1m-ξ1-p+1m+ξ1-pamp1/p∑n=2∞1n-ξ1-q+1n+ξ1-qbnq1/q.For ξ=0, (53) reduces to (2). Hence, (28) is a new extension of (2).


## 5. Conclusions

In this paper, we proposed a new, more accurate extension of Mulholland’s inequality in the whole plane with the best possible constant factor. The equivalent forms, a few particular cases, and the operator expressions were considered and described as some lemmas and theorems in the extension. The method of the real analysis is very important and is the key to prove the equivalent inequalities with the best possible constant factor. The lemmas and theorems can provide an extensive account of this type of inequalities.



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