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}

1_{Faculty of Textile Technology, University of Zagreb, Pierottijeva 6, 10000 Zagreb, Croatia} 2_{Faculty 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, an, bn ≥ 0,such that 0<

_∞

n0a

p

n <∞and 0<

_∞

n0b

q

n <∞, then the following inequality

holdsHardy et al.1:

∞

n0 ∞

m0

ambn mn1 <

π

sinπ/p

_∞

n0

apn

1/p_∞

n0

bqn

1/q

where the constant factorπ/sinπ/pis the best possible. The equivalent form of inequality

1.1issee Yang and Debnath2

∞

n0

_∞

m0

am mn1

p <

π

sinπ/p

p_∞

n0

apn, 1.2

where the constant factorπ/sinπ/ppis 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

mα_nαλ < kαλr

_∞

n1

np1−αλ/r−1apn

1/p_∞

n1

nq1−αλ/s−1bqn

1/q

, 1.3

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

Letφx xαp1−λ/r−1, ϕx xαq1−λ/s−1, ψx xαpλ/s−1 x∈0,∞,and

lp_φ {a{an}∞_n0;a_p,φ:{∞_n0φn|an|p}1/p<∞}.Define a Hilbert-type linear operatorT; for alla∈l_φp,one has

Tan:

∞

m0

lnmα/nα

mαλ_−nαλam. 1.4

Fora∈lp_φ, b∈lqϕ,define the formal inner product ofTaandbas

Ta, b:

∞

n0 ∞

m0

lnmα/nαambn

mαλ_−nαλ . 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< kλsa_p,φb_q,ϕ,

Ta_p,ψ < kλsa_p,φ,

1.6

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:

Ω2

Kx, yfxgydμ1xdμ2

y

≤

Ωϕ

p_xFxfp_xdμ

1x

1/p

Ωψ

q_yGygq_ydμ

2y

1/q ,

1.7

ΩG

1−p_yψ−p_y ΩK

x, yfxdμ1x

p dμ2

y≤

Ωϕ

p_xFxfp_xdμ

1x, 1.8

where

Fx

Ω

Kx, y ψp_y dμ2

y, Gy

Ω

Kx, y

ϕq_x 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. Let1/p1/q 1withp > 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:

Ω2

Kx, yfxgydμ1xdμ2

y

≤

Ωϕ p_xF

1xfpxdμ1x

1/p

Ωψ q_yG

1ygqydμ2y

1/q ,

ΩG

1−p

1

yψ−py ΩK

x, yfxdμ1x

p dμ2

y≤

Ωϕ p_xF

1xfpxdμ1x.

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−λ_{Kx, y is satisfied for every t > 0. Further, we define kα :}

_∞

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

2. Main Results

We applyTheorem 1.2to obtain the following theorem.

Theorem 2.1. Letλ >0, 1/p1/q1withp >1.Let{an}∞n1and{bn}∞n1be two nonnegative real sequences. IfKx, y≥0is homogeneous function of degree−λstrictly decreasing in both parameters

xandy, μ≥0,then the following inequalities hold and are equivalent:

∞

m1 ∞

n1

Kmμ, nμambn

≤L

_∞

m1

mμ1−λpA1−A2_ap m

1/p_∞

n1

nμ1−λqA2−A1_bq n

1/q ,

2.1

∞

n1

nμλ−1p−1pA1−A2

_∞

m1

Kmμ, nμam p

≤Lp

∞

m1

mμ1−λpA1−A2_ap

m, 2.2

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

LkpA2

1/p

k2−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 mμA1_andψnμ nμA2_{in the inequality} 1.7. Then, we have

∞

m1 ∞

n1

Kmμ, nμambn

≤

_∞

m1

mμpA1_Fmμap m

1/p_∞

n1

nμqA2_Gnμbq n

1/q ,

2.4

where Fmμ ∞_n1Kmμ, nμ/nμpA2_and Gnμ ∞

m1Kmμ, n

μ/mμqA1._SinceqA

1 >0 andpA2 >0,the functionsFmμandGnμare strictly

decreasing, where we have

Fmμ< F1

mμ: _∞

0

Kmμ, yμ

yμpA2 dy,

Gnμ< G1

nμ: _∞

0

Kxμ, nμ

xμqA1 dx.

2.5

Using homogeneity of the functionsKand the substitutionu yμ/mμwe get

F1

mμ≤mμ1−λ−pA2

_∞

0

K1, tt−pA2_dtmμ1−λ−pA2_kpA

2

In a similar manner we obtain

G1

nμ≤nμ1−λ−qA1_k2−λ−qA

1

. 2.7

Now, the result follows fromTheorem 1.2.

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

fxp_K

1ϕx−pq, g

yqK2ψ

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∗kpA2

. 2.10

Further, the inequalities2.1and2.2take form

∞

m1 ∞

n1

Kmμ, nμambn

≤L∗

_∞

m1

mμ−1pqA1_ap m

1/p_∞

n1

nμ−1pqA2_bq n

1/q ,

2.11

∞

n1

nμp−11−pqA2

_∞

m1

Kmμ, nμam p

≤L∗p∞ m1

mμ−1pqA1_ap

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

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

nμ1ε < M

1μ−1−ε

_∞

1

xμ−1−εdx

M

ε1με

ε

1μ1

.

2.14

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

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

1≥

∞

n1

∞

1

Kxμ, nμxμ−qA1−ε/p_dxnμ−pA2−ε/q

≥

_∞

1

xμ−qA1−ε/p

_∞

1

Kxμ, yμyμ−pA2−ε/q_dy

dx

_∞

1

xμ−1−ε

_xμ/1μ

0

K1, tt−qA1ε/q_{dt dx}

1

ε1με

1

0

K1, tt−qA1ε/q_dt

_∞

1

xμ−1−ε

_xμ/1μ

1

K1, tt−qA1ε/q_{dt dx}

1

ε1με

1

0

K1, tt−qA1ε/q_dt

_∞

1

K1, tt−qA1−ε/p_{dt .}

2.15

From2.11,2.14, and2.15we get

M

ε

1μ1

≥

1

0

K1, tt−qA1ε/q_dt

_∞

1

K1, tt−qA1−ε/p_{dt. 2.16}

By lettingε → 0,we obtain

M≥

₁

0

K1, tt−qA1_dt

_∞

1

K1, tt−qA1_dtkqA

1

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/xα_yαλ_{is 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

ambn

mμαnμαλ

≤L1

_∞

m1

mμ−1pqA1_ap m

1/p_∞

n1

nμ−1pqA2_bq n

1/q ,

∞

n1

nμp−11−pqA2

⎛ ⎝∞

m1

am

mμαnμαλ

⎞ ⎠

p ≤Lp₁

∞

m1

mμ−1pqA1 apm,

2.18

where the constant factorsL1 1/αB1−pA2/α,1−qA1/αandLp₁are the best possible.

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

become

∞

m1 ∞

n1

ambn

mn2μλ ≤L1

_∞

m1

mμ1−λapm

1/p_∞

n1

nμ1−λbnq

1/q ,

∞

n1

nμp−1λ−1

_∞

m1

am

mn2μλ p

≤Lp₁

∞

m1

mμ1−λapm,

2.19

where the constant factorsL1B1/pλ−1/q,1/qλ−1/p, andLp₁are 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/mμλ_−nμλ_{.Note that the parametersA}

1, A2satisfy conditionpA2qA12−λ.

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

−1, Kx, y xλ−1_yλ−1_/xλ_yλ_{,nonnegative real sequencesa{a}

m}∞m1, b{bm}∞m1,

and the parametersA1A21/pq,we have

∞

m1 ∞

n1

mμλ−1nμλ−1

mμλnμλ ambn≤L2apbq,

∞

n1

_∞

m1

mμλ−1nμλ−1

mμλnμλ am p

≤Lp₂app,

2.20

where the constantsL2 π/λ1/sinπ/p 1/sinπ/qandLp₂ 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, nambn≤L∗

_∞

m1

m−1pqA1_ap m

1/p_∞

n1

n−1pqA2_bq n

1/q ,

∞

n1

np−11−pqA2

_∞

m1

Km, nam p

≤L∗p∞ m1

m−1pqA1_ap 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/xαyαλ,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.