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volume 7, issue 3, article 113, 2006.

Received 05 January, 2006;

accepted 10 April, 2006.

Communicated by:B. Yang

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Journal of Inequalities in Pure and Applied Mathematics

BEST GENERALIZATION OF A HILBERT TYPE INEQUALITY

BAOJU SUN

Zhejiang Water Conservancy & Hydropower College Hangzhou, Zhejiang 310018

People’s Republic Of China.

EMail:[email protected]

c

2000Victoria University ISSN (electronic): 1443-5756 007-06

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Best Generalization of a Hilbert Type Inequality

Baoju Sun

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J. Ineq. Pure and Appl. Math. 7(3) Art. 111, 2006

http://jipam.vu.edu.au

Abstract

By introducing a parameterλ, we have given generalization of Hilbert’s type integral inequality with a best possible constant factor. Also its equivalent form is considered, and the generalized formula corresponding to the double series inequalities are built.

2000 Mathematics Subject Classification:26D15.

Key words: Hilbert’s type integral inequality; Weight coefficient; Weight function;

Hölder’s inequality.

1. Introduction

If p > 1, ¹_p + ¹_q = 1, f, g ≥ 0, satisfy 0 < R∞

0 f^p(t)dt < ∞ and 0 <

R∞

0 g^q(t)dt <∞,then (1.1)

Z ∞

0

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

,

where the constant factor π/(sinπ/p) is the best possible. Inequality (1.1) is called Hardy-Hilbert’s inequality (see [1]) and is important in analysis and applications (cf. Mitrinovi´c et al. [2]). Recently, Yang gave an extension of (1.1) as (see [4,5]):

(1.2) Z ∞

0

Z ∞

0

f(x)g(y) (x+y)^λdxdy

< B

p+λ−2

p ,q+λ−2 q

Z ∞

0

t^1−λf^p(t)dt

¹_pZ ∞

0

t^1−λg^q(t)dt ¹_q

,

where the constant factor B

p+λ−2

p ,^q+λ−2_q

(λ > 2−min{p, q}) is the best possible, and B(u, v) is the β function. Hardy et al. [1] gave an inequality similar to (1.1) as:

(1.3)

Z ∞

0

Z ∞

0

f(x)g(y)

max{x, y}dxdy < pq Z ∞

0

f^p(t)dt

¹_pZ ∞

0

g^q(t)dt ¹_q

,

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where the constant factor pq is the best possible. The double series inequality is:

(1.4)

∞

X

n=1

∞

X

m=1

a_mb_n

max{m, n} < pq

∞

X

n=1

a^p_n

!¹_p _∞ X

m=1

b^q_m

!¹_q ,

where the constant factorpqis the best possible. In particular, ifp=q= 2, one has the Hilbert type integral inequality:

(1.5)

Z ∞

0

Z ∞

0

f(x)g(y)

max{x, y}dxdy <4 Z ∞

0

f²(t)dt Z ∞

0

g²(t)dt ¹₂

.

Recently, Kuang gave an extension of (1.4) as (see [3]):

(1.6)

∞

X

n=1

∞

X

m=1

a_mb_n max{m, n}

<

∞

X

n=1

[pq−G(p, n)]a^p_n

!¹_p _∞ X

m=1

[pq−G(q, n)]b^p_m

!¹_q ,

where G(r, n) = ^r+1/3r−4/3

(2n+1)^1/r > 0 (r = p, q). Yang and Debnath have also considered other Hilbert type integral inequalities in [6].

The main objective of this paper is to build a new inequality with a best constant factor, related to the double integral R∞

0

R∞ 0

f(x)g(y)

max{x^λ,y^λ}dxdy, which improves inequality (1.5). The equivalent form and the corresponding double series form are considered.

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2. Main Results

Theorem 2.1. If λ > 0, p > 1, ¹_p + ¹_q = 1, f, g ≥ 0 such that 0 <

R∞

0 t^p−1−λf^p(t)dt <∞,0<R∞

0 t^q−1−λg^q(t)dt <∞, then one has

(2.1) Z ∞

0

Z ∞

0

f(x)g(y)

max{x^λ, y^λ}dxdy

< pq λ

Z ∞

0

t^p−1−λf^p(t)dt

¹_pZ ∞

0

t^q−1−λg^q(t)dt ¹_q

and (2.2)

Z ∞

0

y^{λ(p−1)−1} Z ∞

0

f(x) max{x^λ, y^λ}dx

p

<pq λ

pZ ∞

0

x^p−1−λf^p(x)dx,

where the constant factors ^pq_λ, ^pq_λp

are the best possible. Inequality (2.1) is equivalent to (2.2). In particular, forλ= 1, (2.1) and (2.2) respectively reduce to the following two equivalent inequalities:

(2.3) Z ∞

0

Z ∞

0

f(x)g(y) max{x, y}dxdy

< pq Z ∞

0

t^p−2f^p(t)dt

¹_pZ ∞

0

t^q−2f^q(t)dt ¹_q

and (2.4)

Z ∞

0

y^p−2 Z ∞

0

f(x) max{x, y}dx

p

dy <(pq)^p Z ∞

0

x^p−2f^p(x)dx.

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Proof. By Hölder’s inequality, one has Z ∞

0

Z ∞

0

f(x)g(y)

max{x^λ, y^λ}dxdy (2.5)

= Z ∞

0

Z ∞

0

"

f(x) (max{x^λ, y^λ})¹^p

y x

_pq^λ⁻¹_p x¹^q⁻¹^p

× g(y) (max{x^λ, y^λ})¹^q

x y

_pq^λ−¹_q

y¹^p⁻¹^q

# dxdy

≤ Z ∞

0

Z ∞

0

f^p(x) max{x^λ, y^λ}

y x

^λ_q⁻¹

x^p^q⁻¹dxdy _p¹

× (Z ∞

0

Z ∞

0

g^q(y) max{x^λ, y^λ}

x y

^λ_p⁻¹

y^q^p⁻¹dxdy )¹_q

. Equality holds in (2.5) if there are two constantsA, B, such thatA²+B² 6= 0 and

A f^p(x) max{x^λ, y^λ}

y x

^λ_q−1

x^p^q⁻¹ =B g^q(y) max{x^λ, y^λ}

x y

^λ_p−1

y^q^p⁻¹

a.e. in (0,∞) × (0,∞), or Ax^p−λf^p(x) = By^q−λg^q(y) =constant a.e. in (0,∞)×(0,∞), this contradicts the fact that 0 < R∞

0 t^p−1−λf^p(t)dt < ∞.

Thus the inequality (2.5) is strict.

Define the weight functionwλ(r, t)as:

w_λ(r, t) = t^λ−1 Z ∞

0

1 max{t^λ, u^λ}

u t

^λ_r−1

du, r =p, q;t ∈(0,∞).

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By computing, one has:

(2.6) w_λ(p, t) = pq

λ =w_λ(q, t), and we obtain

Z ∞

0

Z ∞

0

f(x)g(y)

max{x^λ, y^λ}dxdy

<

Z ∞

0

w_λ(q, t)t^p−1−λf^p(t)dt

¹_pZ ∞

0

w_λ(p, t)t^q−1−λf^q(t)dt ¹_q

= pq λ

Z ∞

0

t^p−1−λf^p(t)dt

¹_pZ ∞

0

t^q−1−λf^q(t)dt ¹_q

.

For0 < ε < λ, settingf(t),e eg(t)as: t ∈ (0,1), f(t) =e eg(t) = 0;t ∈ [1,∞), fe(t) = t^λ−p−ε^p , eg(t) =t^λ−q−ε^q

Z ∞

0

Z ∞

0

fe(x)eg(y)

= Z ∞

1

y

λ−q−ε q

Z ∞

1

max{x^λ, y^λ}x

λ−p−ε

p dx

dy

= Z ∞

1

y

λ−q−ε q

Z y

1

1 y^λx

λ−p−ε

p dx

dy +

Z ∞

1

y

λ−q−ε q

Z ∞

y

1 x^λx

λ−p−ε

p dx

dy

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= p

λ−ε 1

ε + q

λ−λq−ε

+ 1

ε · p

λp−λ+ε

= 1 ε

p

λ−ε + p

λp−λ+ε

− pq

(λ−ε)(λq−λ+ε). On the other hand,

(2.8)

Z ∞

0

t^p−1−λfe^p(t)dt

¹_pZ ∞

0

t^q−1−λeg^q(t)dt ¹_q

= 1 ε.

If the constant factor ^pq_λ in (2.1) is not the best possible, then there exists a positive numberk(withk < ^pq_λ), such that (2.1) is still valid if one replaces ^pq_λ byk. By (2.7) and (2.8), one has:

p

λ−ε + p

λp−λ+ε − pqε

(λ−ε)(λq−λ+ε) (2.9)

=ε Z ∞

0

Z ∞

0

fe(x)fe(y)

max{x^λ, y^λ}dxdy

< εk Z ∞

0

t^p−1−λfe^p(t)dt

¹_pZ ∞

0

t^q−1−λeg^q(t)dt ¹_q

=k.

Settingε → 0⁺, then _λ^p +_(p−1)λ^p ≤ k or ^pq_λ ≤ k. By this contradiction we can conclude that the constant factor ^pq_λ in (2.1) is the best possible.

Define the weight functiong(y)as:

g(y) =y^{λ(p−1)−1} Z ∞

0

p−1

, y∈(0,∞).

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By (2.1), one has:

0<

Z ∞

0

y^q−1−λg^q(y)dy p

(2.10)

= Z ∞

0

y^{λ(p−1)−1} Z ∞

0

f(x) max{x^λ, y^λ}

p

dy p

= Z ∞

0

Z ∞

0

f(x)g(y)

max{x^λ, y^λ}dxdy p

≤pq λ

pZ ∞

0

x^p−1−λf^p(x)dx Z ∞

0

y^q−1−λg^q(y)dy p−1

,

0<

Z ∞

0

y^q−1−λg^q(y)dy (2.11)

= Z ∞

0

y^{λ(p−1)−1} Z ∞

0

p

dy

≤pq λ

pZ ∞

0

x^p−1−λf^p(x)dx <∞.

Hence by using (2.1), (2.10) takes the form of strict inequality; so does (2.11).

One then has (2.2). On other hand, if (2.2) holds, by Hölder’s inequality, one has

Z ∞

0

Z ∞

0

f(x)g(y)

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= Z ∞

0

y^λ+1−q^q

Z ∞

0

h

y^q−1−λ^q g(y)i dy

≤ Z ∞

0

y^{λ(p−1)−1} Z ∞

0

f(x) max{x^λ, y^λ}

p¹_pZ ∞

0

y^q−1−λg^q(y)dy ¹_q

By (2.2), we have (2.1).

If the constant factor in(2.2) is not the best possible, we may show that the constant factor in (2.1) is not the best possible by using (2.12). This is a contradiction. Hence the constant factor in (2.2) is the best possible. Inequality (2.1) is equivalent to (2.2). Thus the theorem is proved.

Remark 1. For p = q = 2, (2.3) reduces to (1.5); (2.1), (2.3) are generaliza- tions of (1.5), but (2.1) is not a generalization of (1.3).

Theorem 2.2. If p > 1, 1/p+ 1/q = 1,0 < λ ≤ min{p, q},a_n ≥ 0, b_n ≥ 0 such that0<

∞

P

n=1

n^p−1−λa^p_n<∞,0<

∞

P

n=1

n^q−1−λa^q_n<∞, one has:

(2.13)

∞

X

n=1

∞

X

m=1

a_mb_n

max{m^λ, n^λ} < pq λ

∞

X

n=1

n^p−1−λa^p_n

!_p¹ _∞ X

n=1

n^q−1−λb^q_n

!¹_q

and (2.14)

∞

X

n=1

n^{λ(p−1)−1}

∞

X

m=1

a_m max{m^λ, n^λ}

!p

<pq λ

p ∞

X

n=1

n^p−1−λa^p_n, where the constant factors ^pq_λ, ^pq_λp

are the best possible. Inequality(2.13)is equivalent to(2.14).

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In particular, forλ = 1,(2.13)and(2.14)respectively reduce to the follow- ing two equivalent inequalities:

(2.15)

∞

X

n=1

∞

X

m=1

a_mb_n

max{m, n} < pq

∞

X

n=1

n^p−2a^p_n

!_p¹ _∞ X

n=1

n^q−2b^q_n

!¹_q

and (2.16)

∞

X

n=1

n^p−2

∞

X

m=1

a_m max{m, n}

!p

<(pq)^p

∞

X

n=1

n^p−2a^p_n.

Proof. Define the weight coefficientwe_λ(λ, n)as:

(2.17) we_λ(r, n) = n^λ−1

∞

X

m=1

1 max{m^λ, n^λ}

m n

^λ_r−1

, r =p, q; n∈N.

Since0< λ≤min{p, q}, we have:

weλ(r, n)< n^λ−1

∞

X

m=1

Z m

m−1

1 max{u^λ, n^λ}

u n

^λ_r−1

du (2.18)

=n^λ−1 Z ∞

0

1 max{u^λ, n^λ}

u n

^λ_r−1

du

=wλ(r, n) = pq

λ, r =p, q

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By Hölder’s inequality and (2.17), following the method of proof in Theorem 2.1, one has:

∞

X

n=1

∞

X

m=1

a_mb_n

max{m^λ, n^λ} <

∞

X

n=1

we_λ(q, n)n^p−1−λa^p_n

!¹_p _∞ X

m=1

we_λ(p, n)n^q−1−λa^p_n

!¹_q

By (2.18) we have (2.13), for0 < ε < λ, settingea_n = n^λ−p−ε^p ,eb_n = n^λ−q−ε^q , n ∈N. Since0< λ≤min{p, q}, by (2.9), we get:

∞

X

n=1

∞

X

m=1

ea_meb_n

max{m^λ, n^λ} >

Z ∞

1

Z ∞

1

f(x)e eg(y)

max{x^λ, y^λ}dxdy

= 1 ε

p

λ−ε + p

λp−λ+ε

− pq

(λ−ε)(λq−λ+ε). On other hand,

∞

X

n=1

n^p−1−λea^p_n

!¹_p _∞ X

m=1

n^q−1−λeb^q_n

!¹_q

=

∞

X

n=1

1

n^1+ε <1 + 1 ε.

By using the above inequalities and the method of proof in Theorem 2.1, we may show that the constant factor in (2.13) is the best possible.

Settingb_nas:

bn =n^{λ(p−1)−1}

∞

X

m=1

a_m max{m^λ, n^λ}

!^p−1 ,

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we obtain:

∞

X

n=1

n^q−1−λb^q_n=

∞

X

n=1

n^{λ(p−1)−1}

∞

X

m=1

a_m max{m^λ, n^λ}

!p

=

∞

X

m=1

∞

X

n=1

a_mb_n max{m^λ, n^λ}.

By (2.13) and using the same method of Theorem2.1, we have (2.14). We may show that the constant factor in (2.14)is the best possible, and inequality (2.13) is equivalent to (2.14).

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References

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

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

[3] J. KUANG AND L. DEBNATH, On new generalizations of Hilbert’s in- equality and their applications, Math. Anal. Appl., 245(1) (2000), 248–265.

[4] B. YANG, On a generalization of Hardy Hilbert’s integral inequality with a best value, Chinese Ann. of Math., A21(4) (2000), 401–408.

[5] B. YANG, On Hardy Hilbert’s integral inequality, J. Math. Anal. Appl., 261 (2001), 295–306.

[6] B. YANG ANDL. DEBNATH, On a new strengthened Hardy-Hilbert’s in- equality, Internat. J. Math. Sci., 21(1) (1998), 403–408.