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volume 7, issue 4, article 153, 2006.

Received 07 February, 2006;

accepted 13 April, 2006.

Communicated by:B. Yang

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Journal of Inequalities in Pure and Applied Mathematics

BEST GENERALIZATION OF HARDY-HILBERT’S INEQUALITY WITH MULTI-PARAMETERS

DONGMEI XIN

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

EMail:xdm77108@gdei.edu.cn

c

2000Victoria University ISSN (electronic): 1443-5756 034-06

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Best Generalization of Hardy-Hilbert’s Inequality With

Multi-Parameters Dongmei Xin

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Abstract

By introducing some parameters and theβfunction and improving the weight function, we obtain a generalization of Hilbert’s integral inequality with the best constant factor. As its applications, we build its equivalent form and some particular results.

2000 Mathematics Subject Classification:26D15.

Key words: Hardy-Hilbert’s inequality, Hölder’s inequality, weight function, βfunc- tion.

1. Introduction

Ifp > 1, ¹_p+¹_q = 1, f, gare non-negative functions such that0<R∞

0 f^p(t)dt <

∞and0<R∞

0 g^q(t)dt <∞, then we have (1.1)

Z ∞ 0

f(x)g(y)

x+y dxdy < π sin(^π_p)

Z ∞ 0

f^p(t)dt

¹_pZ ∞ 0

g^q(t)dt ¹_q

;

(1.2)

Z ∞ 0

f(x) x+ydx

p

dy <

"

π sin(^π_p)

#p

Z ∞ 0

f^p(t)dt, where the constant factors _sin(π/p)^π and

h π sin(π/p)

ip

are the best possible (see [1]).

Inequality (1.1) is well known as Hardy-Hilbert’s integral inequality, which is important in analysis and applications (see [2]). Inequality (1.1) is equivalent to (1.2).

In 2002, Yang [3] gave some generalizations of (1.1) and (1.2) by introducing a parameterλ >0as:

(1.3) Z ∞

0

Z ∞ 0

f(x)g(y) x^λ+y^λ dxdy

< π λsin(^π_p)

Z ∞ 0

t^{(p−1)(1−λ)}f^p(t)dt _p¹

× Z ∞

0

t^{(q−1)(1−λ)}g^q(t)dt ¹_q

;

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(1.4)

Z ∞ 0

y^λ−1 Z ∞

0

f(x) x^λ+y^λdx

p

dy

<

"

π λsin(^π_p)

#p

Z ∞ 0

t^{(p−1)(1−λ)}f^p(t)dt,

where the constant factors _λ_sin(π/p)^π andh

π λsin(π/p)

ip

are the best possible. In- equality (1.3) is equivalent to (1.4).

Whenλ = 1, both (1.3) and (1.4) change to (1.1) and (1.2). Yang [4] gave another generalization of (1.1) by introducing a parameterλand aβfunction.

In 2004, by introducing some parameters and estimating the weight function, Yang [5] gave some extensions of (1.1) and (1.2) with the best constant factors as:

(1.5) Z ∞

0

Z ∞ 0

f(x)g(y) x^λ+y^λ dxdy

< π λsin(^π_r)

Z ∞ 0

x^p(1−^λ^r⁾⁻¹f^p(x)dx

¹_p Z ∞ 0

x^q(1−^λ^s⁾⁻¹g^q(x)dx ¹_q

;

(1.6)

Z ∞ 0

y^pλ^s⁻¹ Z ∞

0

f(x) x^λ+y^λdx

p

dy

<

π λsin(^π_r)

pZ ∞ 0

x^p(1−^λ^r⁾⁻¹f^p(x)dx,

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where the constant factors _λ_sin(π/r)^π andh

π λsin(π/r)

ip

are the best possible. In- equality (1.5) is equivalent to (1.6). Recently, [6, 7, 8, 9] considered some multiple extensions of (1.1).

Under the same conditions with (1.1), we still have (see [1, Th. 342]):

(1.7) Z ∞

0

Z ∞ 0

ln

x y

f(x)g(y) x−y dxdy

<

"

π sin(^π_p)

#2

Z ∞ 0

f^p(t)dt

¹_pZ ∞ 0

g^q(t)dt ¹_q

;

(1.8)

Z ∞ 0



 ln

x y

f(x) x−y dx





p

dy <

"

π sin(^π_p)

#2p

Z ∞ 0

f^p(t)dt,

where the constant factorsh

π sin(π/p)

i2

andh

π sin(π/p)

i2p

are the best possible. In- equality (1.7) is equivalent to (1.8). In recent years, by introducing a parameter λ, Kuang [10] gave an new extension of (1.7).

In 2003, by introducing a parameterλ > 0and the weight function, Yang [11] gave another generalisation of (1.7) and the extended equivalent form as:

(1.9) Z ∞

0

Z ∞ 0

ln

x y

f(x)g(y) x^λ−y^λ dxdy

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<

"

π λsin(^π_p)

#2

Z ∞ 0

t^{(p−1)(1−λ)}f^p(t)dt

¹_pZ ∞ 0

t^{(q−1)(1−λ)}g^q(t)dt ¹_q

;

(1.10)

Z ∞ 0

y^λ−1



 ln

x y

f(x) x^λ−y^λ dx





p

dy

<

"

π λsin(^π_p)

#2p

Z ∞ 0

t^{(p−1)(1−λ)}f^p(t)dt,

where the constant factors h

π λsin(π/p)

i2

and h

π λsin(π/p)

i2p

are the best possible.

Inequality (1.9) is equivalent to (1.10).

In this paper, by using the β function and obtaining the expression of the weight function, we give a new extension of (1.7) with some parameters as (1.5). As applications, we also consider the equivalent form and some other particular results.

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2. Some Lemmas

Lemma 2.1. If p > 1, ¹_p + ¹_q = 1, r > 1, ¹_s + ¹_r = 1, λ > 0,define the weight functionω_λ(s, p, x)as

(2.1) ωλ(s, p, x) = Z ∞

0

ln

x y

x^λ−y^λ ·x^(p−1)(1−^λ^r⁾

y¹⁻^λ^s dy, x∈(0,∞).

Then we have

(2.2) ω_λ(s, p, x) = x^p(1−^λ^r⁾⁻¹ π

λsin(^π_r) 2

.

Proof. For fixedx, settingu= (^y_x)^λ in the integral (2.1) and by [1] (see [1, Th.

342 Remark]), we have ω_λ(s, p, x) = 1

λ² Z ∞

0

lnu

x^λ(u−1)· x^(p−1)(1−^λ^r⁾

(u¹^λx)¹⁻^λ^s xu^λ¹⁻¹du (2.3)

= 1

λ²x^p(1−^λ^r⁾⁻¹ Z ∞

0

lnu

u−1·u⁻¹^rdu

= π

λsin(^π_r) 2

x^p(1−^λ^r⁾⁻¹. Hence, (2.2) is valid and the lemma is proved.

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Note. By (2.3), we still have

ω_λ(r, q, y) = Z ∞

0

ln

x y

x^λ −y^λ ·x^(q−1)(1−^λ^s⁾ y¹⁻^λ^r dx (2.4)

=y^q(1−^λ^s⁾⁻¹ π

λsin(^π_s) 2

.

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3. Main Results and Applications

Theorem 3.1. Ifp > 1, ¹_p+¹_q = 1, r > 1, ¹_s+¹_r = 1, λ >0, f, g ≥0such that 0<R∞

0 x^p(1−^λ^r⁾⁻¹f^p(x)dx <∞,and0<R∞

0 x^q(1−^λ^s⁾⁻¹g^q(x)dx <∞,then we have

(3.1) Z ∞

0

Z ∞ 0

ln

x y

<

π λsin(^π_r)

2Z ∞ 0

x^p(1−^λ^r⁾⁻¹f^p(x)dx ¹_p

× Z ∞

0

, where the constant factorh

π λsin(π/r)

i2

is the best possible. In particular, (a) forr =s= 2,we have

(3.2) Z ∞

0

Z ∞ 0

ln

x y

<

π λ

2Z ∞ 0

x^p(1−^λ²⁾⁻¹f^p(x)dx ¹_p

× Z ∞

0

x^q(1−^λ²⁾⁻¹g^q(x)dx ¹_q

,

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(b) forλ= 1, we have (3.3)

Z ∞ 0

ln

x y

f(x)g(y) x−y dxdy

<

π sin(^π_r)

2Z ∞ 0

x^p^s⁻¹f^p(x)dx

¹_pZ ∞ 0

x^q^r⁻¹g^q(x)dx ¹_q

. Proof. By Hölder’s inequality and Lemma2.1, we have

Z ∞ 0

ln

x y

f(x)g(y) x^λ−y^λ dxdy (3.4)

= Z ∞

0

Z ∞ 0









 ln

x y

x^λ−y^λ





1 p

· x⁽¹⁻^λ^r^)/q y⁽¹⁻^λ^s^)/pf(x)







×









 ln

x y

x^λ−y^λ





1 q

· y⁽¹⁻^λ^s^)/p x⁽¹⁻^λ^r^)/qg(y)





 dxdy

≤





 Z ∞

0



 Z ∞

0

ln x

y

x^λ−y^λ · x^(p−1)(1−^λ^r⁾ y⁽¹⁻^λ^s⁾ dy



f^p(x)dx







1 p

×



Z ∞



Z ∞ ln x

y

x^λ−y^λ · y^(q−1)(1−^λ^s⁾

(1−^λ) dx



g^q(y)dy





1 q

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= Z ∞

0

ω_λ(s, p, x)f^p(x)dx

¹_pZ ∞ 0

ω_λ(r, q, y)g^q(y)dy ¹_q

.

If (3.4) takes the form of equality, then there exist constantsAandB, such that they are not all zero and (see [12])

A ln

x y

x^λ−y^λ ·x^(p−1)(1−^λ^r⁾

y¹⁻^λ^s f^p(x) = B ln

x y

x^λ−y^λ · y^(q−1)(1−^λ^s⁾ x¹⁻^λ^r g^q(y), a.e. in (0,∞)×(0,∞).

We find thatAx·x^p(1−^λ^r⁾⁻¹f^p(x) =By·y^q(1−^λ^s⁾⁻¹g^q(y),a.e. in(0,∞)×(0,∞).

Hence there exists a constantC, such that

Ax·x^p(1−^λ^r⁾⁻¹f^p(x) = C=By·y^q(1−^λ^s⁾⁻¹g^q(y), a.e. in (0,∞).

Without loss of generality, suppose A 6= 0, we may get x^p(1−^λ^r⁾⁻¹f^p(x) = C/(Ax), a.e. in (0,∞), which contradicts 0 < R∞

0 x^p(1−^λ^r⁾⁻¹f^p(x)dx < ∞.

Hence (3.4) takes strict inequality as follows:

(3.5) Z ∞

0

Z ∞ 0

ln

x y

<

Z ∞ 0

ω_λ(s, p, x)f^p(x)dx

¹_pZ ∞ 0

ω_λ(r, q, y)g^q(y)dy ¹_q

. In view of (2.2) and (2.4), we have (3.1).

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If the constant factor h π

λsin(π/r)

i2

in (3.1) is not the best possible, then there exists a positive constantK

withK <h

π λsin(π/r)

i2

and ana >0. We have

(3.6) Z ∞

a

Z ∞ 0

ln

x y

< K Z ∞

a

¹_p Z ∞ a

. Forε >0small enough ε < ^pλ_r

and0< b < a,settingf_εandg_εas:

f_ε(x) =g_ε(x) = 0, x∈(0, b);

f_ε=x⁻¹⁻^ε^p⁺^λ^r, g_ε=x⁻¹⁻^ε^q⁺^λ^s, x∈[b,∞), then we find

(3.7) Z ∞

a

Z ∞ b

ln x

y

fε(x)·gε(y) x^λ−y^λ dxdy

= Z ∞

a

Z ∞ b

ln

x y

x⁻¹⁻^ε^p⁺^λ^r ·y⁻¹⁻^ε^q⁺^λ^s x^λ−y^λ dxdy.

In (3.7), forb→0⁺,by (3.6), we have 1 Z ∞

lnu

u⁻¹⁺¹^s⁻^qλ^ε du=ε

Z ∞Z ∞ln

x y

f_ε(x)g_ε(y)

dxdy≤ K .

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Forε⁺→0,by [1] (see [1, Th. 342 Remark]), it follows that h π

λsin(π/r)

i2

≤K, which contradicts the fact that K < h

π λsin(π/r)

i2

. Hence the constant factor h π

λsin(π/r)

i2

in (3.1) is the best possible. The theorem is proved.

Theorem 3.2. Ifp > 1, ¹_p + ¹_q = 1, r > 1, ¹_s + ¹_r = 1, λ >0, f ≥ 0such that 0<R∞

0 x^p(1−^λ^r⁾⁻¹f^p(x)dx <∞, then we have (3.8)

Z ∞ 0

y^pλ^s⁻¹



 Z ∞

0

ln

x y

f(x) x^λ−y^λ dx





p

dy

<

π λsin(^π_r)

2pZ ∞ 0

x^p(1−^λ^r⁾⁻¹f^p(x)dx,

where the constant h π

λsin(π/r)

i2p

is the best possible. Inequality (3.8) is equiva- lent to (3.1). In particular,

(a) forr =s= 2, we have

(3.9)

Z ∞ 0

y^pλ² ⁻¹



 Z ∞

0

ln

x y

f(x) x^λ−y^λ dx





p

dy

<π λ

2pZ ∞ 0

x^p(1−^λ²⁾⁻¹f^p(x)dx,

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(b) forλ= 1, we have

(3.10)

Z ∞ 0

y^p^s⁻¹



 Z ∞

0

ln x

y

f(x) x−y dx





p

dy

<

π sin(^π_r)

2pZ ∞ 0

x^p^s⁻¹f^p(x)dx.

Proof. Setting a real functiong(y)as

g(y) = y^pλ^s ⁻¹



 Z ∞

0

ln

x y

f(x) x^λ−y^λ dx





p−1

, y∈(0,∞), then by (3.1), we find

Z ∞ 0

y^q(1−^λ^s⁾⁻¹g^q(y)dy p

(3.11)

=





 Z ∞

0

y^pλ^s ⁻¹



 Z ∞

0

ln x

y

f(x) x^λ−y^λ dx





p

dy







p

=



 Z ∞

0

Z ∞ 0

ln

x y

f(x)g(y) x^λ −y^λ dxdy





p

≤

π 2pZ ∞

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× Z ∞

0

x^q(1−^λ^s⁾⁻¹g^q(x)dx p−1

. Hence we obtain

(3.12) 0<

Z ∞ 0

y^q(1−^λ^s⁾⁻¹g^q(y)dy

≤

π λsin(^π_r)

2pZ ∞ 0

x^p(1−^λ^r⁾⁻¹f^p(x)dx <∞.

By (3.1), both (3.11) and (3.12) take the form of strict inequality, and we have (3.8).

On the other hand, suppose that (3.8) is valid. By Hölder’s inequality, we find

Z ∞ 0

ln

x y

f(x)g(y) x^λ−y^λ dxdy (3.13)

= Z ∞

0



y^λ^s⁻¹^p Z ∞

0

ln

x y

f(x) x^λ−y^λ dx



 h

y⁻^λ^s⁺¹^pg(y)i dy

≤





 Z ∞

0

y^pλ^s ⁻¹



 Z ∞

0

ln

x y

f(x) x^λ−y^λ dx





p

dy







1 p

× Z ∞

0

y^q(1−^λ^s⁾⁻¹g^q(y)dy ¹_q

.

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Then by (3.8), we have (3.1). Hence (3.1) and (3.8) are equivalent.

If the constant h π

λsin(π/r)

i2p

in (3.8) is not the best possible, by using (3.13), we may get a contradiction that the constant factor in (3.1) is not the best possible. Thus we complete the proof of the theorem.

Remark 1.

(a) For r = q, s = p,Inequality (3.1) reduces to (1.9) and (3.8) reduces to (1.10).

(b) Inequality (3.1) is an extension of (1.7) with parameters(λ, r, s).

(c) It is interesting that inequalities (1.9) and (3.2) are different, although they have the same parameters and possess a best constant factor.

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References

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

[2] 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.

[3] BICHENG YANG, On an extension of Hardy-Hilbert’s inequality, J.

Math. Anal. Appl., 23A(2) (2002), 247–254.

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

Appl., 261 (2001), 295–306.

[5] BICHENG YANG, On the extended Hilbert’s integral inequality and ap- plications, International Review of Pure and Applied Mathematics, (2005), 10–11.

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

[7] LEPING HE, JIANGMING YU AND MINZHE GAO, An extension of Hilbert’s integral inequality, J. Shaoguan Univ., (Natural Science), 23(3) (2002), 25–30.

[8] JICHANG KUANG, On new extensions of Hilbert’s integral inequality, J.

Math. Anal. Appl., 235 (1999), 608–614.

[9] M. KRNI ´CANDJ. PE ˇCARI ´C, General Hilbert’s and Hardy’s inequalities, Math. Inequal. & Applics., 8(1) (2005), 29–51.

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[10] JICHANG KUANG AND L. DEBNATH, On new generalizations of Hilbert’s inequality and their applications, J. Math. Anal. Appl., 245 (2000), 248–265.

[11] BICHENG YANG, On a generalization of a Hilbert’s type integral inequal- ity and its applications, Mathematics Applications, 16(2) (2003), 82–86.

[12] JICHANG KUANG, Applied Inequalities, Shandong Science and Tech- nology Press, Jiinan, (2004).