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volume 6, issue 2, article 39, 2005.

Received 04 June, 2004;

accepted 18 March, 2005.

Communicated by:B.G. Pachpatte

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Journal of Inequalities in Pure and Applied Mathematics

ON A NEW MULTIPLE EXTENSION OF HILBERT’S INTEGRAL INEQUALITY

BICHENG YANG

Department of Mathematics, Guangdong Institute of Education, Guangzhou, Guangdong 510303, People’s Republic Of China.

EMail:bcyang@pub.guangzhou.gd.cn URL:http://page.gdei.edu.cn/~yangbicheng

c

2000Victoria University ISSN (electronic): 1443-5756 114-04

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On a New Multiple Extension of Hilbert’s Integral Inequality

Bicheng Yang

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J. Ineq. Pure and Appl. Math. 6(2) Art. 39, 2005

http://jipam.vu.edu.au

Abstract

This paper gives a new multiple extension of Hilbert’s integral inequality with a best constant factor, by introducing a parameterλand theΓfunction. Some particular results are obtained.

2000 Mathematics Subject Classification:26D15.

Key words: Hilbert’s integral inequality; Weight coefficient ,Γfunction.

1. Introduction

Iff, g≥0satisfy 0<

Z ∞ 0

f²(x)dx <∞ and 0<

Z ∞ 0

g²(x)dx <∞,

then (1.1)

Z ∞ 0

f(x)g(y)

x+y dxdy < π Z ∞

0

f²(x)dx Z ∞

0

g²(x)dx ¹₂

, where the constant factorπis the best possible (cf. Hardy et al. [2]). Inequality (1.1) is well known as Hilbert’s integral inequality, which had been extended by Hardy [1] as:

Ifp > 1, ¹_p +¹_q = 1, f, g ≥0satisfy 0<

Z ∞ 0

f^p(x)dx <∞ and 0<

Z ∞ 0

g^q(x)dx <∞,

then (1.2)

Z ∞ 0

f(x)g(y) x+y dxdy

< π sin

π p

Z ∞

0

f^p(x)dx

¹_pZ ∞ 0

g^q(x)dx ¹_q

,

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where the constant factor _sin(π/p)^π is the best possible. Inequality (1.2) is called Hardy- Hilbert’s integral inequality, and is important in analysis and its appli- cations (cf. Mitrinovi´c et al.[6]).

Recently, by introducing a parameterλ,Yang [9] gave an extension of (1.2) as:

Ifλ >2−min{p, q}, f, g ≥0satisfy 0<

Z ∞ 0

x^1−λf^p(x)dx <∞ and 0<

Z ∞ 0

x^1−λg^q(x)dx <∞,

then (1.3)

Z ∞ 0

f(x)g(y) (x+y)^λdxdy

< k_λ(p) Z ∞

0

x^1−λf^p(x)dx

¹_p Z ∞ 0

x^1−λg^q(x)dx ¹_q

,

where the constant factork_λ(p) = B

p+λ−2

p ,^q+λ−2_q

is the best possible (B(u, v) is theβ function). Forλ= 1,inequality (1.3) reduces to (1.2).

On the problem for multiple extension of (1.1), [3,4] gave some new results and Yang [8] gave an improvement of their works as:

Ifn ∈N\{1}, p_i >1,Pn i=1

1

pi = 1, λ > n− min

1≤i≤n{p_i}, f_i ≥0,satisfy 0<

Z ∞ 0

x^n−1−λf_i^pⁱ(x)dx <∞ (i= 1,2, . . . , n),

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then (1.4)

Z ∞ 0

· · · Z ∞

0

1 Pn

j=1x_jλ n

Y

i=1

f_i(x_i)dx₁. . . dx_n

< 1 Γ(λ)

n

Y

i=1

Γ

p_i+λ−n p_i

Z ∞ 0

x^n−1−λf_i^pⁱ(x)dx ¹

pi ,

where the constant factor _Γ(λ)¹ Qn i=1Γ

pi+λ−n pi

is the best possible. Forn= 2, inequality (1.4) reduces to (1.3). It follows that (1.4) is a multiple extension of (1.3), (1.2) and (1.1).

In 2003, Yang et. al [11] provided an extensive account of the above results.

The main objective of this paper is to build a new extension of (1.1) with a best constant factor other than (1.4), and give some new particular results. That is

Theorem 1.1. Ifn ∈N\{1}, p_i >1,Pn i=1

1

pi = 1, λ > 0, f_i ≥0,satisfy 0<

Z ∞ 0

x^pⁱ^−1−λf_i^pⁱ(x)dx <∞ (i= 1,2, . . . , n),

then (1.5)

Z ∞ 0

· · · Z ∞

0

1 Pn

j=1x_jλ n

Y

i=1

f_i(x_i)dx₁. . . dx_n

< 1 Γ(λ)

n

Y

i=1

Γ λ

p_i

Z ∞ 0

x^pⁱ^−1−λf_i^pⁱ(x)dx ¹

pi ,

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where the constant factor _Γ(λ)¹ Qn i=1Γ

λ pi

is the best possible. In particular, (a) forλ= 1,we have

(1.6) Z ∞

0

· · · Z ∞

0

Qn

i=1f_i(x_i) Pn

j=1xj

dx₁. . . dx_n

<

n

Y

i=1

Γ 1

p_i

Z ∞ 0

x^pⁱ⁻²f_i^pⁱ(x)dx ¹

pi ;

(b) for n = 2, using the symbol of (1.3) and setting ekλ(p) = B λ

p,^λ_q

,we have

(1.7) Z ∞

0

Z ∞ 0

f(x)g(y) (x+y)^λdxdy

<ek_λ(p) Z ∞

0

x^p−1−λf^p(x)dx

¹_p Z ∞ 0

x^q−1−λg^q(x)dx ¹_q

, where the constant factors in (1.6) and (1.7) are still the best possible.

In order to prove the theorem, we introduce some lemmas.

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2. Some Lemmas

Lemma 2.1. Ifk ∈N, r_i >1 (i= 1,2, . . . , k+ 1),andPk+1

i=1 r_i =λ(k),then (2.1)

Z ∞ 0

· · · Z ∞

0

1

1 +Pk

j=1u_jλ(k) k

Y

j=1

u^r_j^j⁻¹du₁. . . du_k = Qk+1

i=1 Γ(r_i) Γ(λ(k)) .

Proof. We establish (2.1) by mathematical induction. Fork= 1, sincer1+r2 = λ(1),and (see [7])

(2.2) B(p, q) = Z ∞

0

u^p−1

(1 +u)^p+qdu=B(q, p) (p, q >0),

we have (2.1). Suppose fork ∈ N,that (2.1) is valid. Then for k+ 1, since r₁+Pk+1

i=2 r_i =λ(k+ 1), by settingv =u₁.

1 +Pk+1 j=2 u_j

, we obtain Z ∞

0

· · · Z ∞

0

1

1 +Pk+1 j=1uj

λ(k+1) k+1

Y

j=1

u^r_j^j⁻¹du₁. . . du_k+1 (2.3)

= Z ∞

0

· · · Z ∞

0

v^r¹⁻¹

1 +Pk+1 j=2u_jr1

Qk+1 j=2u^r_j^j⁻¹

1 +Pk+1

j=2 u_jλ(k+1)

(1 +v)^λ(k+1)

dvdu₂. . . du_k+1

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= Z ∞

0

· · · Z ∞

0

1 +Pk+1

j=2u_j^λ(k+1)−r1du₂. . . du_k+1

× Z ∞

0

v^r¹⁻¹

(1 +v)^λ(k+1)dv.

In view of (2.2) and the assumption ofk, we have (2.4)

Z ∞ 0

v^r¹⁻¹

(1 +v)^λ(k+1)dv= 1

Γ(λ(k+ 1))Γ

k+1

X

i=2

r_i

! Γ(r₁);

(2.5) Z ∞

0

· · · Z ∞

0

1 +Pk+1

j=2u_jλ(k+1)−r1du₂. . . du_k+1 =

Qk+2 i=2 Γ(r_i) Γ

Pk+1 i=2 r_i. Then, by (2.5), (2.4) and (2.3), we find

Z ∞ 0

· · · Z ∞

0

Qk+1 j=1 u^r_j^j⁻¹

1 +Pk+1 j=1uj

λ(k+1)du₁. . . du_k+1 = Qk+2

i=1 Γ(ri) Γ(λ(k+ 1)).

Hence (2.1) is valid fork ∈Nby induction. The lemma is proved.

Lemma 2.2. Ifn ∈ N\{1}, p_i >1 (i = 1,2, . . . , n),Pn i=1

1

pi = 1andλ > 0, set the weight coefficientω(x_i)as

(2.6) ω(x_i) :=x

λ pj

i

Z ∞ 0

· · · Z ∞

0

Qn

j=1(j6=i)x^(λ−p_j ^j^)/p^j Pn

j=1x_jλ dx₁. . . dxi−1dx_i+1. . . dx_n.

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Then, eachω(x_i)is constant, that is (2.7) ω(x_i) = 1

Γ(λ)

n

Y

j=1

Γ λ

pj

, (i= 1,2, . . . , n).

Proof. Fixi. Settingpe_n =p_i,andpe_j =p_j, u_j =x_j/x_i, forj = 1,2, . . . , i−1;

pe_j =p_j+1, u_j =x_j+1/x_i, forj =i, i+ 1, . . . , n−1in (2.6), by simplification, we have

(2.8) ω(x_i) = Z ∞

0

· · · Z ∞

0

1

1 +Pn−1 j=1 u_jλ

n−1

Y

j=1

u

−1+^λ

pj

j du₁. . . dun−1.

Substitution of n −1 fork, λ forλ(k)and λ/pe_j for r_j (j = 1,2, . . . , n)into (2.1), in view of (2.8), we have

ω(x_i) = 1 Γ(λ)

n

Y

j=1

Γ λ

pe_j

= 1

Γ(λ)

n

Y

j=1

Γ λ

p_j

.

Hence, (2.7) is valid. The lemma is proved.

Lemma 2.3. As in the assumption of Lemma2.2, for0< ε < λ, we have

I :=ε Z ∞

1

· · · Z ∞

1

Qn

i=1x^(λ−p_i ⁱ^−ε)/pⁱ Pn

j=1x_jλ dx₁. . . dx_n

≥ 1

Γ(λ)

n

Y

i=1

Γ λ

p_i

(ε →0⁺).

(2.9)

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Proof. Settingu_i =x_i/x_n(i= 1,2, . . . , n−1)in the following, we find

I =ε Z ∞

1

x^−1−ε_n





 Z ∞

x⁻¹n

· · · Z ∞

x⁻¹n

Qn−1

i=1 u^(λ−p_i ⁱ^−ε)/pⁱ

1 +Pn−1

j=1 u_jλdu1. . . dun−1





dxn

≥ε Z ∞

1

x^−1−ε_n





 Z ∞

0

· · · Z ∞

0

Qn−1

i=1 u^(λ−p_i ⁱ^−ε)/pⁱ

1 +Pn−1

j=1 u_jλdu1. . . dun−1





dxn

−ε Z ∞

1

x⁻¹_n

n−1

X

j=1

A_j(x_n)dx_n, (2.10)

where, forj = 1,2, . . . , n−1, A_j(x_n)is defined by (2.11) A_j(x_n) :=

Z

· · · Z

Dj

Qn−1

i=1 u^(λ−p_i ⁱ^−ε)/pⁱ (1 +Pn−1

j=1 u_j)^λ du₁. . . du_n−1,

satisfyingD_j ={(u₁, u₂, . . . , un−1)|0< u_j ≤x⁻¹_n , 0< u_k <∞(k 6=j)}.

Without loss of generality, we estimate the integralA_j(x_n)forj = 1.

(a) Forn= 2, we have A₁(x_n) =

Z x⁻¹n

0

1

(1 +u₁)^λu^(λ−p₁ ¹^−ε)/p¹du₁

≤ Z x⁻¹n

0

u^(λ−p₁ ¹^−ε)/p¹du₁ = p₁

λ−εx^{−(λ−ε)/p}_n ¹;

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(b) Forn∈N\{1,2},by (2.1), we have

A₁(x_n)≤ Z ∞

0

· · · Z ∞

0

Qn−1 i=2 u⁻¹⁺

λ−ε pi

i

1 +Pn−1

j=2 u_jλdu₁. . . dun−1

Z x⁻¹n

0

u

λ−p1−ε p1

1 du₁

≤ p₁x⁻

λ−ε p1

n

λ−ε Z ∞

0

· · · Z ∞

0

Qn−1

i=2 u^{−1+(λ−ε)/p}_i ⁱ

1 +Pn−1

j=2u_j^{(λ−ε)(1−p}⁻¹₁ )du₁. . . dun−1

= p₁x⁻

λ−ε p1

n

λ−ε ·

Qn

i=2Γ(^λ−ε_p

i ) Γ((λ−ε)(1−p⁻¹₁ )).

By virtue of the results of (a) and (b), forj = 1,2, . . . , n−1,we have (2.12) A_j(x_n)≤ p_j

λ−εx^{−(λ−ε)/p}_n ^jO_j(1) (ε→0⁺, n∈N\{1}).

By (2.11), since forε→0⁺, Z ∞

0

· · · Z ∞

0

Qn−1

i=1 u^{−1+(λ−ε)/p}_i ⁱ

1 +Pn−1 j=1 uj

λ du₁. . . du_n−1 = Qn

i=1Γ

λ pi

Γ(λ) +o(1);

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Z ∞ 1

x⁻¹_n

n−1

X

j=1

A_j(x_n)dx_n =

n−1

X

j=1

Z ∞ 1

x⁻¹_n A_j(x_n)dx_n

≤

n−1

X

j=1

p_j

λ−εOj(1) Z ∞

1

x^{−1−(λ−ε)/p}_n ^jdxn

=

n−1

X

j=1

p_j λ−ε

2

O_j(1),

then by (2.10) , we find

I ≥



 Qn

i=1Γ

λ pi

Γ(λ) +o(1)



−ε

n−1

X

j=1

p_j λ−ε

2

O_j(1)

→ Qn

i=1Γ

λ pi

Γ(λ) (ε→0⁺).

Thereby, (2.9) is valid and the lemma is proved.

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3. Proof of the Theorem and Remarks

Proof of Theorem1.1. By Hölder’s inequality, we have

(3.1) J :=

Z ∞ 0

· · · Z ∞

0

1 Pn

j=1x_jλ n

Y

i=1

fi(xi)dx1. . . dxn

= Z ∞

0

· · · Z ∞

0











n

Y

i=1

f_i(x_i) Pn

j=1x_jλ/pi







x^(pⁱ^−λ)(1−p

−1 i ) i

n

Y

j=1

(j6=i)

x

λ−pj pj

j







1 pi











dx₁. . . dx_n

≤

n

Y

i=1









 Z ∞

0

· · · Z ∞

0

f_i^pⁱ(x_i) Pn

j=1x_jλx^(pⁱ^−λ)(1−p

−1 i ) i

n

Y

j=1

(j6=i)

x

λ−pj pj

j dx₁. . . dx_n











1 pi

.

If (3.1) takes the form of equality, then there exists constantsC1, C2, . . . , Cn, such that they are not all zero and for anyi6=k ∈ {1,2, . . . , n}(see [5]),

C_if_i^pⁱ(x_i)x^(pⁱ^−λ)(1−p

−1 i ) i

Pn

j=1x_jλ

n

Y

j=1

(j6=i)

x

λ−pj pj

j =C_kf_k^p^k(x_k)x^(p^k^−λ)(1−p

−1 k ) k

Pn

j=1x_jλ

n

Y

j=1

(j6=k)

x

λ−pj pj

j ,

a.e. in(0,∞)× · · · ×(0,∞).

(3.2)

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Assume thatC_i 6= 0.By simplification of (3.2), we find x^p_iⁱ^−λf_i^pⁱ(x_i) =F(x₁, . . . , xi−1, x_i+1, . . . , x_n)

=constant a.e. in(0,∞)× · · · ×(0,∞), which contradicts the fact that0 <R∞

0 x^p_iⁱ^−λ−1f_i^pⁱ(x)dx < ∞.Hence by (2.6) and (3.1), we conclude

(3.3) J <

n

Y

i=1

Z ∞ 0

ω(x_i)x^p_iⁱ^−1−λf_i^pⁱ(x_i)dx_i _pi¹

. Then by (2.7), we have (1.5).

For0< ε < λ,settingfei(xi)as:fei(xi) = 0,forxi ∈(0,1);

fe_i(x_i) =x^(λ−p_i ⁱ^−ε)/pⁱ, forx_i ∈[1,∞) (i= 1,2, . . . , n), then we find

(3.4) ε

n

Y

i=1

Z ∞ 0

x^p_iⁱ^−1−λfe_i^pⁱ(x_i)dx_i _pi¹

= 1.

By (2.9), we find (3.5) ε

Z ∞ 0

· · · Z ∞

0

1 Pn

j=1x_jλ n

Y

i=1

fe_i(x_i)dx₁. . . dx_n

=I ≥ 1 Γ(λ)

n

Y

i=1

Γ λ

p_i

(ε→0⁺).

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If the constant factor _Γ(λ)¹ Qn i=1Γ

λ pi

in (1.5) is not the best possible, then there exists a positive constant K < _Γ(λ)¹ Qn

i=1Γ

λ pi

, such that (1.5) is still valid if one replaces _Γ(λ)¹ Qn

i=1Γ

λ pi

byK.In particular, one has

ε Z ∞

0

· · · Z ∞

0

1 Pn

j=1x_jλ n

Y

i=1

fe_i(x_i)dx₁. . . dx_n

< εK

n

Y

i=1

Z ∞ 0

x^pⁱ^−1−λfe_i^pⁱ(x)dx _pi¹

,

and in view of (3.4) and (3.5), it follows that _Γ(λ)¹ Qn i=1Γ

λ pi

≤ K (ε →

0⁺).This contradicts the factK < _Γ(λ)¹ Qn i=1Γ

λ pi

.Hence the constant factor

1 Γ(λ)

Qn i=1Γ

λ pi

in (1.5) is the best possible.

The theorem is proved.

Remark 1. Forλ= 1, inequality (1.7) reduces to (see [10])

(3.6) Z ∞

0

Z ∞ 0

f(x)g(y) x+y dxdy

< π sin

π p

Z ∞

0

x^p−2f^p(x)dx

¹_pZ ∞ 0

x^q−2g^q(x)dx ¹_q

.

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Forp =q = 2, both (3.6) and (1.2) reduce to (1.1). It follows that inequalities (3.6) and (1.2) are different extensions of (1.1). Hence, inequality (1.5) is a new multiple extension of (1.1). Since all the constant factors in the obtained inequalities are the best possible, we have obtained new results.

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References

[1] G.H. HARDY, Note on a theorem of Hilbert concerning series of positive terms, Proc. Math. Soc., 23(2) (1925), Records of Proc. XLV-XLVI.

[2] G.H. HARDY, J.E. LITTLEWOOD AND G. POLYA, Inequalities, Cam- bridge University Press, Cambridge, 1952.

[3] YONG HONG, All-sided generalization about Hardy-Hilbert’s integral in- equalities, Acta Math. Sinica, 44(4) (2001), 619–626.

[4] LEPING HE, JIANGMING YU AND MINGZHE GAO, An extension of Hilbert’s integral inequality, J. Shaoguan University (Natural Science), 23(3) (YEAR), 25–29.

[5] JICHANG KUANG, Applied Inequalities, Hunan Education Press, Changsha, 2004.

[6] D.S. MITRINOVI ´C, J.E. PE ˇCARI ´CANDA.M. FINK, Inequalities Involv- ing Functions and Their Integrals and Derivatives, Kluwer Academic Pub- lishers, Boston, 1991.

[7] ZHUXI WANG AND DUNRIN GUO, An Introduction to Special Func- tions, Science Press, Beijing, 1979.

[8] BICHENG YANG, On a multiple Hardy-Hilbert’s integral inequality, Chi- nese Annals of Math., 24A(6) (2003), 743–750.

[9] BICHENG YANG, On Hardy-Hilbert’s integral inequality, J. Math. Anal.

Appl., 261 (2001), 295–306.

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[10] BICHENG YANG, On the extended Hilbert’s integral inequality, J. In- equal. Pure and Appl. Math., 5(4) (2004), Art. 96. [ONLINE: http:

//jipam.vu.edu.au/article.php?sid=451]

[11] BICHENG YANG AND Th. M. RASSIAS, On the way of weight coeffi- cient and research for the Hilbert-type inequalities, Math. Ineq. Appl., 6(4) (2003), 625–658.