# On a Parametric Mulholland-Type Inequality and Applications

Authors: Bicheng Yang; Meifa Huang; Yanru Zhong
Journal: Abstract and Applied Analysis
 (2019)

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## Abstract

In this paper, by the use of the weight functions, and the idea of introducing parameters, a discrete Mulholland-type inequality with the general homogeneous kernel and the equivalent form are given. The equivalent statements of the best possible constant factor related to a few parameters are provided. As applications, the operator expressions and a few particular examples are considered.


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## Body

## 1. Introduction

Assuming that0<∑m=1∞am2<∞and0<∑n=1∞bn2<∞, we have the following discrete Hilbert’s inequality with the best possible constant factor π (cf. [1], Theorem 315):(1)∑m=1∞∑n=1∞ambnm+n<π∑m=1∞am2∑n=1∞bn21/2.We still have the following Mulholland’s inequality with the same best possible constant π (cf. [1], Theorem 343): (2)∑m=2∞∑n=2∞ambnln⁡mn<π∑m=2∞mam2∑n=2∞nbn21/2.If 0<∫0∞f2(x)dx<∞ and 0<∫0∞g2(y)dy<∞, then we have the following Hilbert’s integral inequality:(3)∫0∞∫0∞fxgyx+ydxdy<π∫0∞f2xdx∫0∞g2ydy1/2,with the best possible constant factor π (cf. [1], Theorem 316).Inequalities (1), (2), and (3) and their extensions with the conjugate exponents (p,q)(p>1,1/p+1/q=1) and independent parameters are important in analysis and its applications (cf. [2–13]).The following half-discrete Hilbert-type inequality was provided (cf. [1], Theorem 351). If K(x)(x>0) is decreasing, p>1,1/p+1/q=1,0<ϕ(s)=∫0∞K(x)xs-1dx<∞, then(4)∫0∞xp-2∑n=1∞Knxanpdx<ϕp1q∑n=1∞anp.Some new extensions of (4) were provided by [14–19].In 2016, by the use of the technique of real analysis, Hong [20] considered some equivalent statements of the extensions of (1) with the best possible constant factor related to a few parameters. The other similar works about the extensions of (3) were provided by [21–25].In this paper, according to the way given by [20], by the use of the weight functions and the idea of introducing parameters, a discrete Mulholland-type inequality with the general homogeneous kernel and the equivalent form are given, which is an extension of (2). The equivalent statements of the best possible constant factor related to a few parameters are provided. As applications, the operator expressions and a few particular examples are considered.


## 2. Some Lemmas

In what follows, we suppose thatp>1,1/p+1/q=1,λ∈R,λi,λ-λi≤1(i=1,2),  kλ(x,y) is a positive homogeneous function of degree-λ, satisfying, for any u,x,y>0,(5)kλux,uy=u-λkλx,y.Also, kλ(x,y) is decreasing with respect to x,y>0 (or ∂/∂xkλx,y≤0, ∂/∂ykλx,y≤0x,y>0), such that, for γ=λ1,λ-λ2,(6)kλγ≔∫0∞kλu,1uγ-1du∈R+=0,∞.We still assume that am,bn≥0(m,n∈N∖{1}={2,3,…}), satisfying (7)0<∑m=2∞lnp1-λ-λ2/p+λ1/q-1⁡mm1-pamp<∞and0<∑n=2∞lnq1-λ2/p+λ-λ1/q-1⁡nn1-pbnq<∞.Definition 1.
Define the following weight functions:(8)ωλλ2,m≔lnλ-λ2⁡m∑n=2∞kλln⁡m,ln⁡nlnλ2-1⁡nnm∈N∖1,(9)ϖλλ1,n≔lnλ-λ1⁡n∑m=2∞kλln⁡m,ln⁡nlnλ1-1mmn∈N∖1.Lemma 2.
We have the following inequalities:(10)ωλλ2,m<kλλ-λ2m∈N∖1,(11)ϖλλ1,n<kλλ1n∈N∖1.Proof.
For λ 2 - 1 ≤ 0, it is evident that kλ(ln⁡m,ln⁡t)lnλ2-1⁡t/t is a strictly decreasing function with respect to t>1. By the decreasing property, setting u=ln⁡m/ln⁡t, it follows that(12)ωλλ2,m<lnλ-λ2⁡m∫1∞kλln⁡m,ln⁡tlnλ2-1⁡ttdt=∫0∞kλu,1uλ-λ2-1du=kλλ-λ2.Hence, we have (10).Forλ1-1≤0, it is evident that kλ(ln⁡t,ln⁡n)lnλ1-1⁡t/t is a strictly decreasing function with respect to t>1. By the decreasing property, setting u=ln⁡t/ln⁡n, we find that(13)ϖλλ1,n<lnλ-λ1⁡n∫1∞kλln⁡t,ln⁡nlnλ1-1⁡ttdt=∫0∞kλu,1uλ1-1du=kλλ1.Hence, we have (11).Lemma 3.
We have the following inequality:(14)I≔∑n=2∞∑m=2∞kλln⁡m,ln⁡nambn<kλ1/pλ-λ2kλ1/qλ1∑m=2∞lnp1-λ-λ2/p+λ1/q-1⁡mm1-pamp1/p∑n=2∞lnq1-λ-λ1/q+λ2/p-1⁡nn1-qbnq1/q.Proof.
By Hölder’s inequality with weight (cf. [26]), we obtain(15)I≔∑n=2∞∑m=2∞kλln⁡m,ln⁡nlnλ2-1p⁡nn1/pln1-λ1/q⁡mm-1/qam×lnλ1-1/q⁡mm1/qln1-λ2/p⁡nn-1/pbn≤∑m=2∞∑n=2∞kλln⁡m,ln⁡nlnλ2-1⁡nnlnp-11-λ1⁡mm1-pamp1/p×∑n=2∞∑m=2∞kλln⁡m,ln⁡nlnλ1-1⁡mmlnq-11-λ2-1⁡nn1-qbnq1/q=∑m=2∞ωλλ2,mlnp1-λ-λ2/p+λ1/q-1⁡mm1-pamp1/p×∑n=2∞ϖλλ1,nlnq1-λ-λ1/q+λ2/p-1⁡nn1-qbnq1/q.Then by (10) and (11), we have (14).Remark 4.
By (14), for λ1+λ2=λ, we find(16)0<∑m=2∞lnp1-λ1-1⁡mm1-pamp<∞and0<∑n=2∞lnq1-λ2-1⁡nn1-pbnq<∞,and the following inequality:(17)∑n=2∞∑m=2∞kλln⁡m,ln⁡nambn<kλλ1∑m=2∞lnp1-λ1-1⁡mm1-pamp1/p∑n=2∞lnq1-λ2-1⁡nn1-qbnq1/q.In particular, for p=q=2, we have(18)∑n=2∞∑m=2∞kλln⁡m,ln⁡nambn<kλλ1∑m=2∞mln2λ1-1⁡mam2∑n=2∞nln2λ2-1⁡nbn21/2.For λ=1, k1(x,y)=1/x+y, λ1=λ2=1/2, (18) reduces to (2). Hence, (17) is an extension of (18) and (2).Lemma 5.
The constant factork(λ1) in (17) is the best possible.Proof.
For anyε>0, we set(19)a~m≔lnλ1-ε/p-1⁡mm,b~n≔lnλ2-ε/q-1⁡nnm,n∈N∖1.If there exists a constant M(M≤kλ(λ1)), such that (17) is valid when replacing kλ(λ1) by M, then, in particular, we have(20)I~≔∑n=2∞∑m=2∞kλln⁡m,ln⁡na~mb~n<M∑m=2∞lnp1-λ1-1⁡mm1-pa~mp1/p∑n=2∞lnq1-λ2-1⁡nn1-pb~nq1/q.We obtain(21)I~<M∑m=2∞lnp1-λ1-1⁡mm1-plnpλ1-ε-p⁡mmp1/p∑n=2∞lnq1-λ2-1⁡nn1-qlnqλ2-ε-1⁡nnq1/q=Mln-ε-1⁡22+∑m=3∞ln-ε-1⁡mm<Mln-ε-1⁡22+∫2∞ln-ε-1⁡ttdt=Mεlnε⁡2ε2ln⁡2+1.By the decreasing property and Fubini theorem (cf. [27]), we find(22)I~=∑n=2∞∑m=2∞kλln⁡m,ln⁡nlnλ1-1⁡mmlnε/p⁡m·lnλ2-1⁡nnlnε/q⁡n≥∫2∞∫2∞kλln⁡x,ln⁡ylnλ1-ε/p-1⁡xx·lnλ2-ε/q-1⁡yydxdyu=ln⁡xln⁡y=∫2∞ln-ε-1⁡yy∫ln⁡2/ln⁡y∞kλu,1uλ1-ε/p-1dudy=∫2∞ln-ε-1⁡yy∫ln⁡2/ln⁡y1kλu,1uλ1-ε/p-1dudy+∫2∞ln-ε-1⁡yy∫1∞kλu,1uλ1-ε/p-1dudy=∫01∫η+21/u∞ln-ε-1⁡yydykλu,1uλ1-ε/p-1du+1εlnε⁡2∫1∞kλu,1uλ1-ε/p-1du=1εlnε⁡2∫01kλu,1uλ1+ε/q-1du+∫1∞kλu,1uλ1-ε/p-1du.Then we have(23)∫01kλu,1uλ1+ε/q-1du+∫1∞kλu,1uλ1-ε/p-1du<Mε2ln⁡2+1.For ε→0+, by Fatou lemma (cf. [27]), we find(24)kλλ1=∫01lim_ε→0+kλu,1uλ1+ε/q-1du+∫1∞lim_ε→0+kλu,1uλ1-ε/p-1du]≤lim_ε→0+∫01kλu,1uλ1+ε/q-1du+∫1∞kλu,1uλ1-ε/p-1du≤M.Hence, M=kλ(λ1) is the best possible constant factor of (17).Remark 6.
Settingλ^1≔λ-λ2/p+λ1/q, λ^2≔λ-λ1/q+λ2/p, we find(25)λ^1+λ^2=λ-λ2p+λ1q+λ-λ1q+λ2p=λp+λq=λ,λ^1≤1p+1q=1,λ^2≤1q+1p=1,and by Hölder’s inequality (cf. [26]), we have (26)0<kλλ-λ^2=kλλ^1=kλλ-λ2p+λ1q=∫0∞kλu,1uλ-λ2/p+λ1/q-1du=∫0∞kλu,1uλ-λ2-1/puλ1-1/qdu≤∫0∞kλu,1uλ-λ2-1du1/p∫0∞kλu,1uλ1-1du1/q=kλ1/pλ-λ2kλ1/qλ1<∞.We can rewrite (14) as follows:(27)I<kλ1/pλ-λ2kλ1/qλ1∑m=2∞lnp1-λ^1-1⁡mm1-pamp1/p∑n=2∞lnq1-λ^2-1⁡nn1-qbnq1/q.Lemma 7.
If the constant factorkλ1/p(λ-λ2)kλ1/q(λ1) in (14) is the best possible, then λ1+λ2=λ.Proof.
If the constant factorkλ1/p(λ-λ2)kλ1/q(λ1) in (14) is the best possible, then, by (27) and (17), the unique best possible constant factor must be kλ(λ^1)(∈R+), namely,(28)kλλ^1=kλ1/pλ-λ2kλ1/qλ1.We observe that (26) keeps the form of equality if and only if there exist constantsAandB, such that they are not all zero and (cf. [26])(29)Auλ-λ2-1=Buλ1-1a.e.inR+=0,∞.Assuming that A≠0 (otherwise, B=A=0), it follows that uλ-λ2-λ1=B/Aa.e.in R+, and then λ-λ2-λ1=0, namely, λ1+λ2=λ.


## 3. Main Results

Theorem 8.
Inequality (14) is equivalent to (30)J≔∑n=2∞lnpλ-λ1/q+λ2/p-1⁡nn∑m=2∞kλln⁡m,ln⁡namp1/p<kλ1/pλ-λ2kλ1/qλ1∑m=2∞lnp1-λ-λ2/p+λ1/q-1⁡mm1-pamp1/p.If the constant factor in (14) is the best possible, then so is the constant factor in (30).Proof.
Suppose that (30) is valid. By Hölder’s inequality (cf. [26]), we find (31)I=∑n=2∞ln-1/p+λ-λ1/q+λ2/p⁡nn1/p∑m=2∞kλln⁡m,ln⁡namln1/p-λ-λ1/q+λ2/p⁡nn-1/pbn≤J∑n=2∞lnq1-λ-λ1/q+λ2/p-1⁡nn1-qbnq1/q.Then by (30), we obtain (14).
On the other hand, assuming that (14) is valid, we set(32)bn≔lnpλ-λ1/q+λ2/p-1⁡nn∑m=2∞kλln⁡m,ln⁡namp-1,n∈N∖1.If J=0, then (30) is naturally valid; if J=∞, then it is impossible to make (30) valid, namely, J<∞. Suppose that 0<J<∞. By (14), it follows that(33)∑n=2∞lnq1-λ-λ1/q+λ2/p-1⁡nn1-qbnq=Jp=I<kλ1/pλ-λ2kλ1/qλ1∑m=2∞lnp1-λ-λ2/p+λ1/q-1⁡mm1-pamp1/p∑n=2∞lnq1-λ-λ1/q+λ2/p-1nn1-qbnq1/q,J=∑n=2∞lnq1-λ-λ1/q+λ2/p-1⁡nn1-qbnq1/p<kλ1/pλ-λ2kλ1/qλ1∑m=2∞lnp1-λ-λ2/p+λ1/q-1⁡mm1-pamp1/p,namely, (30) follows, which is equivalent to (14).
If the constant factor in (14) is the best possible, then so is constant factor in (30). Otherwise, by (31), we would reach a contradiction that the constant factor in (14) is not the best possible.Theorem 9.
The statements (i), (ii), (iii), and (iv) are equivalent as follows:(i)
kλ1/p(λ-λ2)kλ1/q(λ1) is independent of p,q(ii)
kλ1/p(λ-λ2)kλ1/q(λ1) is expressible as a single integral(iii)
kλ1/p(λ-λ2)kλ1/q(λ1) is the best possible constant factor of (14)(iv)
λ1+λ2=λIf the statement (iv) follows, namely,λ1+λ2=λ, then we have (17) and the following equivalent inequality with the best possible constant factor kλ(λ1):(34)∑n=2∞lnpλ2-1⁡nn∑m=2∞kλln⁡m,ln⁡namp1/p<kλλ1∑m=2∞lnp1-λ1-1⁡mm1-pamp1/p.Proof.
(i)=>(ii). Sincekλ1/p(λ-λ2)kλ1/q(λ1) is independent of p,q, we find(35)kλ1/pλ-λ2kλ1/qλ1=limp→∞⁡limq→1+⁡kλ1/pλ-λ2kλ1/qλ1=kλλ1,namely, kλ1/p(λ-λ2)kλ1/q(λ1) is expressible as a single integral(36)kλλ1=∫0∞kλu,1uλ1-1du.
(ii)=>(iv). In (26), if kλ1/p(λ-λ2)kλ1/q(λ1) is expressible as a single integral kλ(λ-λ2/p+λ1/q), then (26) keeps the form of equality, which follows that λ1+λ2=λ.
(iv)=>(i). Ifλ1+λ2=λ, then kλ1/p(λ-λ2)kλ1/q(λ1)=kλ(λ1), which is independent of p,q. Hence, we have (i)⇔(ii)⇔(iv).
(iii)=>(iv). By Lemma7, we have λ1+λ2=λ.
(iv)=>(iii). By Lemma5, for λ1+λ2=λ, kλ1/p(λ-λ2)kλ1/q(λ1)(=kλ(λ1)) is the best possible constant factor of (14). Therefore, we have (iii) ⇔(iv).
Hence, the statements (i), (ii), (iii), and (iv) are equivalent.Remark 10.
(i) Forλ=1,λ1=1/q,λ2=1/p in (17) and (34), we have the following equivalent inequalities with the best possible constant factor k1(1/q):(37)∑n=2∞∑m=2∞k1ln⁡m,ln⁡nambn<k11q∑m=2∞ampm1-p1/p∑n=2∞bnqn1-q1/q,(38)∑n=2∞1n∑m=2∞k1ln⁡m,ln⁡namp1/p<k11q∑m=2∞ampm1-p1/p.
(ii) Forλ=1, λ1=1/p, λ2=1/q in (17) and (34), we have the following equivalent inequalities with the best possible constant factor k1(1/p):(39)∑n=2∞∑m=2∞k1ln⁡m,ln⁡nambn<k11p∑m=2∞lnp-2⁡mm1-pamp1/p∑n=2∞lnp-2⁡nn1-qbnq1/q,(40)∑n=2∞lnp-2⁡nn∑m=2∞k1ln⁡m,ln⁡namp1/p<k11p∑m=2∞lnp-2⁡mm1-pamp1/p.
(iii) Forp=q=2, both (37) and (39) reduce to(41)∑n=2∞∑m=2∞k1ln⁡m,ln⁡nambn<k112∑m=2∞mam2∑n=2∞nbn21/2,and both (38) and (40) reduce to the equivalent form of (41) as follows:(42)∑n=2∞1n∑m=2∞k1ln⁡m,ln⁡nam21/2<k112∑m=2∞mam21/2.


## 4. Operator Expressions and Some Particular Cases

We set functions(43)φm≔lnp1-λ-λ2/p+λ1/q-1⁡mm1-p,ψn≔lnq1-λ-λ1/q+λ2/p-1⁡nn1-q,where (44)ψ1-pn=lnpλ-λ1/q+λ2/p-1⁡nnm,n∈N∖1.Define the following real normed spaces:(45)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<∞.Assuming thata∈lp,φ, setting(46)c=cnn=2∞,cn≔∑m=2∞kλln⁡m,ln⁡nam,n∈N∖1,we can rewrite (30) as follows:(47)cp,ψ1-p<kλ1/pλ-λ2kλ1/qλ1ap,φ<∞,namely, c∈lp,ψ1-p.Definition 11.
Define a Mulholland-type operatorT:lp,φ→lp,ψ1-p as follows: for any a∈lp,φ, there exists a unique representation c∈lp,ψ1-p. Define the formal inner product of Ta and b∈lq,ψ and the norm of T as follows:(48)Ta,b≔∑n=2∞∑m=2∞kλln⁡m,ln⁡nambn,T≔supa≠θ∈lp,φ⁡Tap,ψ1-pap,φ.By Theorems 8 and 9, we have the following.Theorem 12.
Ifa∈lp,φ,b∈lq,ψ,ap,φ,bq,ψ>0, then we have the following equivalent inequalities:(49)Ta,b<kλ1/pλ-λ2kλ1/qλ1ap,φbq,ψ,(50)Tap,ψ1-p<kλ1/pλ-λ2kλ1/qλ1ap,φ.Moreover, λ1+λ2=λ if and only if the constant factor kλ1/p(λ-λ2)kλ1/q(λ1)=kλ(λ1) in (49) and (50) is the best possible, namely,(51)T=kλλ1.Example 13.
We setkλ(x,y)≔1/(cx+y)λ(c,λ>0;x,y>0). Then we find kλ(ln⁡m,ln⁡n)=1/lnλ⁡mcn. For 0<λi,λ-λi≤1(i=1,2), kλ(x,y) is a positive homogeneous function of degree -λ, such that kλ(x,y) is decreasing with respect tox,y>0, and for γ=λ1,λ-λ2,(52)kλγ=∫0∞uγ-1cu+1λdu=1cγ∫0∞vγ-1v+1λdv=1cγBγ,λ-γ∈R+.
In view of Theorem12, it follows that λ1+λ2=λ if and only if (53)T=kλλ1=1cλ1Bλ1,λ2.Example 14.
We setkλ(x,y)≔ln⁡(cx/y)/(cx)λ-yλ(c>0,λ>0;x,y>0). Then we find kλ(ln⁡m,ln⁡n)=ln(ln⁡mc/ln⁡n)/lnλ⁡mc-lnλ⁡n. For 0<λi,λ-λi≤1(i=1,2), kλ(x,y) is a positive homogeneous function of degree -λ, such that kλ(x,y) is decreasing with respect tox,y>0 (cf. [2], Example 2.2.1), and for γ=λ1,λ-λ2,(54)kλγ=∫0∞uγ-1ln⁡cucuλ-1du=1cγλ2∫0∞vγ/λ-1ln⁡vv-1dv=1cγπλsin⁡πγ/λ2∈R+.
In view of Theorem12, it follows that λ1+λ2=λ if and only if (55)T=kλλ1=1cλ1πλsin⁡πλ1/λ2.Example 15.
We setkλ(x,y)≔1/∏k=1s(xλ/s+ckyλ/s)(0<c1≤⋯≤cs,λ>0;x,y>0). Then we find (56)kλln⁡m,ln⁡n=1∏k=1slnλ/s⁡m+cklnλ/s⁡n.For 0<λi,λ-λi≤1(i=1,2), kλ(x,y) is a positive homogeneous function of degree -λ, such that kλ(x,y) is decreasing with respect to x,y>0, and for γ=λ1,λ-λ2, byExample 1 of [28], it follows that(57)kλsγ=∫0∞tγ-1∏k=1stλ/s+ckdt=πsλsin⁡πsγ/λ∑k=1scksγ/λ-1∏j=1j≠ks1cj-ck∈R+.
In view of Theorem12, it follows that λ1+λ2=λ if and only if (58)T=kλsλ1=πsλsin⁡πsλ1/λ∑k=1scksλ1/λ-1∏j=1j≠ks1cj-ck.
In particular, forc1=⋯=cs=c, we have kλ(x,y)=1/(xλ/s+cyλ/s)s and(59)T=k~λsλ1≔∫0∞tλ1-1tλ/s+csdt=sλc1-λ1/λs∫0∞vsλ1/λ-1v+1sdv=sλc1-λ1/λsBsλ1λ,sλ2λ.If s=1, then we have kλ(x,y)=1/xλ+cyλ, kλ(ln⁡m,ln⁡n)=1/lnλ⁡m+clnλ⁡n, and(60)T=k~λ1λ1=1λc1-λ1/λπsin⁡πλ1/λ.


## 5. Conclusions

In this paper, by the use of the weight functions and the idea of introducing parameters, a discrete Mulholland-type inequality with the general homogeneous kernel and the equivalent form are given in Lemma3 and Theorem 8. The equivalent statements of the best possible constant factor related to a few parameters are considered in Theorem 9. As applications, the operator expressions and some particular examples are given in Theorem 12 and Examples 13–15. The lemmas and theorems provide an extensive account of this type of inequalities.



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