Hardy-Hilbert-Type Inequalities with a Homogeneous Kernel in Discrete Case

Volume 2010, Article ID 912601,8pages doi:10.1155/2010/912601

Research Article

Hardy-Hilbert-Type Inequalities with a Homogeneous Kernel in Discrete Case

Josip Pe ˇcari´c

and Predrag Vukovi ´c

1Faculty of Textile Technology, University of Zagreb, Pierottijeva 6, 10000 Zagreb, Croatia

2Faculty of Teacher Education, University of Zagreb, Ante Starˇcevi´ca 55, 40000 ˇCakovec, Croatia

Correspondence should be addressed to Predrag Vukovi´c,[email protected] Received 4 September 2009; Accepted 16 February 2010

Academic Editor: Ondˇrej Doˇsl ´y

Copyrightq2010 J. Peˇcari´c and P. Vukovi´c. 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.

The main objective of this paper is a study of some new generalizations of Hilbert’s and Hardy- Hilbert’s type inequalities. We apply our general results to homogeneous functions. We shall obtain, in a similar way as Yang did in2009, that the constant factors are the best possible when the parameters satisfy appropriate conditions.

1. Introduction

Hilbert and Hardy-Hilbert type inequalitiessee1are very significant weight inequalities which play an important role in many fields of mathematics. Although classical, such inequalities have attracted the interest of numerous mathematicians and have been generalized in many different ways. Also the numerous mathematicians reproved them using various techniques. Some possibilities of generalizing such inequalities are, for example, various choices of nonnegative measures, kernels, sets of integration, extension to multidimensional case, and so forth.

Similar inequalities, in operator form, appear in harmonic analysis where one investigates properties of boundedness of such operators. This is the reason why Hilbert’s inequality is so popular and represents field of interest of numerous mathematicians: since Hilbert till nowadays.

We start with the following two discrete inequalities, which are the well-known Hilbert and Hardy-Hilbert type inequalities. More precisely, if p > 1, 1/p 1/q 1, a_n, b_n ≥ 0,such that 0< _∞

n0a^p_n <∞and 0< _∞

n0b^q_n <∞, then the following inequality holdsHardy et al.1:

∞ n0

∞ m0

a_mb_n

mn1 < π sin

π/p _∞

n0

a^p_n

1/p_∞

n0

b^q_n 1/q

, 1.1

where the constant factorπ/sinπ/pis the best possible. The equivalent form of inequality 1.1issee Yang and Debnath2

∞ n0

_∞

m0

a_m mn1

p

<

π sin

π/p p∞

n0

a^p_n, 1.2

where the constant factorπ/sinπ/p^pis still the best possible.

In this paper we refer to a recent paper of Yangsee3. In 2005, Yang3gave some extension of Hilbert’s inequality with two pairs of conjugate exponentsp, q,r, s p, r >1, and two parametersα, λ >0 αλ≤min{r, s}as

∞ m1

∞ n1

ambn

m^αn^αλ < kαλr _∞

n1

n^{p1−αλ/r−1}a^p_n

1/p_∞

n1

n^{q1−αλ/s−1}b^q_n

1/q

, 1.3

where the constant factork_αλr 1/αBλ/r, λ/sis the best possible.

Letφx xα^{p1−λ/r−1}, ϕx xα^{q1−λ/s−1}, ψx xα^pλ/s−1 x∈0,∞,and l^p_φ {a{an}^∞_n0;a_p,φ:{_∞

n0φn|an|^p}^1/p<∞}.Define a Hilbert-type linear operatorT; for alla∈l_φ^p,one has

Tan: ^∞

m0

lnmα/nα

mα^λ−nα^λam. 1.4 Fora∈l^p_φ, b∈l^q_ϕ,define the formal inner product ofTaandbas

Ta, b:^∞

n0

∞ m0

lnmα/nαamb_n

mα^λ−nα^λ . 1.5

Zhongsee4proved the following theorem.

Theorem 1.1. Suppose that p, q and r, s are two pairs of conjugate exponents, r > 1, p >

1, 1/2 ≤ α ≤ 1, 0 < λ ≤ 1, an, bn ≥ 0.If a_p,φ > 0, b_q,ϕ > 0,then one has the equivalent inequalities as

Ta, b< k_λsa_p,φb_q,ϕ, Ta_p,ψ < kλsa_p,φ,

1.6

where the constant factork_λs 1/λB1/s,1/r²is the best possible.

Results in this paper will be based on the following general form of Hilbert’s and Hardy-Hilbert’s inequality proven in 5. All the measures are assumed to be σ-finite on some measure spaceΩ. Let 1/p1/q 1 withp > 1, Kx, y, fx, gy, ϕx, ψybe nonnegative functions. Then the following inequalities hold and are equivalent:

Ω²K x, y

fxg y

dμ1xdμ2

y

≤

Ωϕ^pxFxf^pxdμ1x _1/p

Ωψ^qyGyg^qydμ2y _1/q

,

1.7

ΩG^1−p y

ψ^−p

y

ΩK x, y

fxdμ1x _p

dμ2

y

≤

Ωϕ^pxFxf^pxdμ1x, 1.8

where

Fx

Ω

K x, y ψ^p

y dμ₂ y

, G

y

Ω

K x, y

ϕ^qx dμ₁x. 1.9

It is of great importance to consider the case when the functions Fx and Gy, defined by1.9, are bounded. More precisely, Krni´c and Peˇcari´c in5proved the following result.

Theorem 1.2. Let 1/p1/q 1 withp > 1, Kx, y, fx, gy, ϕx, ψybe nonnegative functions andFx ≤ F₁x, Gy ≤ G₁y,whereFxandGyare defined by1.9. Then the following inequalities hold and are equivalent:

Ω²K x, y

fxg y

dμ₁xdμ2

y

≤

Ωϕ^pxF1xf^pxdμ1x 1/p

Ωψ^qyG1yg^qydμ2y 1/q

,

ΩG^1−p₁ y

ψ^−p

y

ΩK x, y

fxdμ1x p

dμ2

y

≤

Ωϕ^pxF1xf^pxdμ1x.

1.10

In this paper a generalization of Theorem 1.1 for a general type of homogeneous kernels is obtained. Recall that for a homogeneous function Kx, y of degree −λ, λ > 0, equality Ktx, ty t^−λKx, y is satisfied for every t > 0. Further, we define kα : _∞

0K1, tt^−αdtand suppose thatkα<∞for 1−λ < α <1.

In what follows, without further explanation, we assume that all series and integrals exist on the respective domains of their definitions.

2. Main Results

We applyTheorem 1.2to obtain the following theorem.

Theorem 2.1. Letλ >0, 1/p1/q1 withp >1.Let{an}^∞_n1and{bn}^∞_n1be two nonnegative real sequences. IfKx, y≥0 is homogeneous function of degree−λstrictly decreasing in both parameters xandy, μ≥0,then the following inequalities hold and are equivalent:

∞ m1

∞ n1

K

mμ, nμ a_mb_n

≤L _∞

m1

mμ_{1−λpA1−A2} a^p_m

1/p_∞

n1

nμ_{1−λqA2−A1} b^q_n

1/q

,

2.1

∞ n1

nμ_{λ−1p−1pA1−A2∞}

m1

Kmμ, nμam p

≤L^p ∞ m1

mμ_{1−λpA1−A2}

a^p_m, 2.2

whereA₁∈max{1−λ/q,0},1/q, A2∈max{1−λ/p,0},1/pand Lk

pA2

_1/p k

2−λ−qA1

_1/q

. 2.3

Proof. We use the inequalities1.7,1.8, andTheorem 1.2with counting measure. First, we prove the inequality2.1. Putϕmμ mμ^A1andψnμ nμ^A2in the inequality 1.7. Then, we have

∞ m1

∞ n1

K

mμ, nμ a_mb_n

≤ _∞

m1

mμ_pA1 F

mμ a^p_m

1/p_∞

n1

nμ_qA2 G

nμ b^q_n

1/q

,

2.4

where Fmμ _∞

n1Kmμ, nμ/nμ^pA2and Gnμ _∞

m1Kmμ, n μ/mμ^qA1.SinceqA1 >0 andpA2 >0,the functionsFmμandGnμare strictly decreasing, where we have

F mμ

< F₁ mμ

: _∞

0

K

mμ, yμ yμpA2 dy, G

nμ

< G₁ nμ

: _∞

0

K

xμ, nμ xμ_qA1 dx.

2.5

Using homogeneity of the functionsKand the substitutionu yμ/mμwe get

F₁ mμ

≤

mμ_{1−λ−pA2∞}

0

K1, tt^−pA2dt

mμ_1−λ−pA2 k

pA₂

. 2.6

In a similar manner we obtain

G1

nμ

≤

nμ_1−λ−qA1 k

2−λ−qA1

. 2.7

Now, the result follows fromTheorem 1.2.

Remark 2.2. Equality in the previous theorem is possible only if

fx^pK₁ϕx^−pq, g y_q

K₂ψ y_−pq

, 2.8

for arbitrary constantsK1andK2see5. Condition2.8immediately gives that nontrivial case of equality in2.1and2.2leads to divergent series.

Now, we consider some special choice of the parametersA₁ andA₂.More precisely, let the parametersA1andA2satisfy constraint

pA₂qA₁ 2−λ. 2.9

Then, the constantLfromTheorem 2.1becomes

L^∗k pA₂

. 2.10

Further, the inequalities2.1and2.2take form ∞

m1

∞ n1

K

mμ, nμ a_mb_n

≤L^∗ _∞

m1

mμ_−1pqA1 a^p_m

1/p_∞

n1

nμ_−1pqA2 b_n^q

1/q

,

2.11

∞ n1

nμ_{p−11−pqA2∞}

m1

K

mμ, nμ am

p

≤L^∗p∞

m1

mμ_−1pqA1

a^p_m. 2.12

In the following theorem we show, in a similar way as Yang did in6, that if the parameters A₁andA₂ satisfy condition2.9, then one obtains the best possible constant. To prove this result we need the next lemmasee6.

Lemma 2.3. Iffx≥0is decreasing in0,∞and strictly decreasing in a subinterval of0,∞, andI₀:_∞

0 fxdx <∞,then

I1: _∞

1

fxdx≤^∞

n1

fn< I0. 2.13

Theorem 2.4. Letλ, μ, A1, A2, andKx, ybe defined as inTheorem 2.1. If the parametersA1and A₂satisfy conditionpA₂qA₁2−λ,then the constantsL^∗kpA2andL^∗pin the inequalities 2.11and2.12are the best possible.

Proof. For this purpose, withε > 0, setam mμ^{−qA1−ε/p} andbn nμ^{−pA2−ε/q}.Now, let us suppose that there exists a smaller constant 0< M < L^∗such that the inequality2.11 is valid. LetJdenote the right-hand side of2.11. UsingLemma 2.3, we have

JM

1μ_−1−ε ^∞

n2

1

nμ_1ε < M

1μ_−1−ε

_∞

1

xμ_−1−ε dx

M ε

1μ_ε ε

1μ1

.

2.14

Further, letIdenote the left-hand side of the inequality2.11, for above choice of sequences

a_mand b_n.Applying, respectively,Lemma 2.3, Fubini’s theorem, and substitutiont x μ/yμ,we have

1≥^∞

n1

_∞

1

K

xμ, nμ

xμ_{−qA1−ε/p}

dx

nμ_{−pA2−ε/q}

≥ _∞

1

xμ_{−qA1−ε/p∞}

1

K

xμ, yμ

yμ_{−pA2−ε/q} dy

dx

_∞

1

xμ_{−1−εxμ/1μ}

0

K1, tt^−qA1ε/qdt dx 1

ε 1με

₁

0

K1, tt^−qA1ε/qdt

_∞

1

xμ_{−1−εxμ/1μ}

1

K1, tt^−qA1ε/qdt dx

1 ε

1μ_{ε 1}

0

K1, tt^−qA1ε/qdt _∞

1

K1, tt^{−qA1−ε/p}dt .

2.15

From2.11,2.14, and2.15we get

M ε

1μ1

≥ ₁

0

K1, tt^−qA1ε/qdt _∞

1

K1, tt^{−qA1−ε/p}dt. 2.16

By lettingε → 0,we obtain

M≥ ₁

0

K1, tt^−qA1dt _∞

1

K1, tt^−qA1dtk qA₁

. 2.17

Using symmetry of the functionKx, y,we havekqA1 kpA2 L^∗.Now, from2.17 we obtain a contradiction with assumptionM < L^∗kpA2.

Finally, equivalence of the inequalities2.11and2.12means that the constantL^∗p is the best possible in the inequality2.12. This completes the proof.

We proceed with some special homogeneous functions. Since the functionKx, y 1/x^αy^αλis homogeneous of degree−αλ,by usingTheorem 2.4we obtain the following.

Corollary 2.5. Let λ > 0, α > 0, μ ≥ 0.Suppose that the parameters A1, A2 satisfy condition pA₂gA₁2−αλ.Then the following inequalities hold and are equivalent:

∞ m1

∞ n1

a_mb_n mμα

nμαλ

≤L_{1 ∞}

m1

mμ_−1pqA1 a^p_m

1/p_∞

n1

nμ_−1pqA2 b^q_n

1/q

,

∞ n1

nμ_{p−11−pqA2}⎛

⎝^∞

m1

am

mμ_α

nμ_αλ

⎞

⎠

p

≤L^p₁ ∞ m1

mμ_−1pqA1 a^p_m,

2.18

where the constant factorsL1 1/αB1−pA2/α,1−qA1/αandL^p₁are the best possible.

Remark 2.6. If we putα1, A₁A₂ 2−λ/pqinCorollary 2.5, then the inequalities2.18 become

∞ m1

∞ n1

a_mb_n

mn2μλ ≤L_{1 ∞}

m1

mμ_1−λ a^p_m

1/p_∞

n1

nμ_1−λ b_n^q

1/q

,

∞ n1

nμ_{p−1λ−1∞}

m1

a_m mn2μ_λ

p

≤L^p₁ ∞ m1

mμ_1−λ a^pm,

2.19

where the constant factorsL1B1/pλ−1/q,1/qλ−1/p, andL^p₁are the best possible.

For λ 1 we obtain nonweighted case with the best possible constant L₁ B1/p,1/q.

Settingμ1/2 andλ1 in the inequalities2.19we obtain the inequalities1.1and1.2 from Introduction.

Remark 2.7. It is easy to see thatTheorem 2.4is the generalization ofTheorem 1.1. Namely, let us defineA1 1/q−λ/qr, A2 1/p−λ/ps, andKmμ, nμ lnmμ/n μ/mμ^λ−nμ^λ.Note that the parametersA₁, A₂satisfy conditionpA₂qA12−λ.

Then, the best possible constantL^∗fromTheorem 2.4becomeskλsfromTheorem 1.1see also4.

Remark 2.8. Similarly as in Corollary 2.5, for the homogeneous function of degree

−1, Kx, y x^λ−1y^λ−1/x^λy^λ,nonnegative real sequencesa{am}^∞_m1, b{bm}^∞_m1, and the parametersA₁A₂1/pq,we have

∞ m1

∞ n1

mμ_λ−1

nμ_λ−1 mμ_λ

nμ_λ ambn≤L2a_pb_q, ∞

n1

_∞

m1

mμ_λ−1

nμ_λ−1 mμ_λ

nμ_λ a_m

p

≤L^p₂a^pp,

2.20

where the constantsL2 π/λ1/sinπ/p 1/sinπ/qandL^p₂ are the best possible.

Remark 2.9. Let λ, A₁, A₂, and Kx, y be defined as in Theorem 2.1. Take μ 0 in the inequalities2.11and2.12. By usingTheorem 2.4we get equivalent inequalities for general homogeneous kernelKx, y:

∞ m1

∞ n1

Km, namb_n≤L^∗ _∞

m1

m^−1pqA1a^p_m

1/p_∞

n1

n^−1pqA2b_n^q

1/q

,

∞ n1

n^{p−11−pqA2} _∞

m1

Km, nam p

≤L^∗p∞

m1

m^−1pqA1a^p_m,

2.21

where the constant factorsL^∗kpA2andL^∗pare the best possible.

SettingA₁1/q−λ/qr, A₂ 1/p−λ/psin the inequalities2.21we obtain the result from6. Similarly, for above choice of the parametersA₁, A₂, andKx, y 1/x^αy^αλ,we obtain Yang’s result1.3from Introduction.

References

1 G. H. Hardy, J. E. Littlewood, and G. P ´olya, Inequalities, Cambridge University Press, Cambridge, UK, 2nd edition, 1967.

2 B. Yang and L. Debnath, “On a new generalization of Hardy-Hilbert’s inequality and its applications,”

Journal of Mathematical Analysis and Applications, vol. 233, no. 2, pp. 484–497, 1999.

3 B. Yang, “On best extensions of Hardy-Hilbert’s inequality with two parameters,” Journal of Inequalities in Pure and Applied Mathematics, vol. 6, no. 3, article 81, pp. 1–15, 2005.

4 W. Zhong, “A Hilbert-type linear operator with the norm and its applications,” Journal of Inequalities and Applications, vol. 2009, Article ID 494257, 18 pages, 2009.

5 M. Krni´c and J. Peˇcari´c, “General Hilbert’s and Hardy’s inequalities,” Mathematical Inequalities &

Applications, vol. 8, no. 1, pp. 29–52, 2005.

6 B. Yang, “On a Hilbert-type operator with a class of homogeneous kernels,” Journal of Inequalities and Applications, vol. 2009, Article ID 572176, 9 pages, 2009.