By introducing a parameterλ, we have given generalization of Hilbert’s type in- tegral inequality with a best possible constant factor

http://jipam.vu.edu.au/

Volume 7, Issue 3, Article 113, 2006

BEST GENERALIZATION OF A HILBERT TYPE INEQUALITY

BAOJU SUN

ZHEJIANGWATERCONSERVANCY& HYDROPOWERCOLLEGE

HANGZHOU, ZHEJIANG310018 PEOPLE’SREPUBLICOFCHINA

[email protected]

Received 05 January, 2006; accepted 10 April, 2006 Communicated by B. Yang

ABSTRACT. By introducing a parameterλ, we have given generalization of Hilbert’s type in- tegral 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.

Key words and phrases: Hilbert’s type integral inequality; Weight coefficient; Weight function; Hölder’s inequality.

2000 Mathematics Subject Classification. 26D15.

1. INTRODUCTION

Ifp > 1, ¹_p +¹_q = 1, f, g ≥0, satisfy0<R∞

0 f^p(t)dt <∞and0<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 theβ 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

,

ISSN (electronic): 1443-5756

c 2006 Victoria University. All rights reserved.

007-06

where the constant factorpqis 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

!_p¹ _∞ 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.

2. MAINRESULTS

Theorem 2.1. If λ > 0, p > 1, ¹_p + ¹_q = 1, f, g ≥ 0such that0 < 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_p Z ∞

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, for λ = 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

_p¹ Z ∞

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.

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

0

Z ∞

0

f(x)g(y)

max{x^λ, y^λ}dxdy = Z ∞

0

Z ∞

0

"

f(x) (max{x^λ, y^λ})^1p

y x

_pq^λ−_p¹

x^1q−p1 (2.5)

× g(y) (max{x^λ, y^λ})^1q

x y

_pq^λ−¹

q

y^1p−1q

# dxdy

≤ Z ∞

0

Z ∞

0

f^p(x) max{x^λ, y^λ}

y x

^λ_q−1

x^pq−1dxdy ¹_p

× (Z ∞

0

Z ∞

0

g^q(y) max{x^λ, y^λ}

x y

^λ_p−1

y^pq−1dxdy )¹_q

.

Equality holds in (2.5) if there are two constantsA,B, such thatA²+B² 6= 0and A f^p(x)

max{x^λ, y^λ} y

x ^λ_q−1

x^pq−1 =B g^q(y) max{x^λ, y^λ}

x y

^λ_p−1

y^qp−1

a.e. in(0,∞)×(0,∞), orAx^p−λf^p(x) =By^q−λg^q(y) =constant a.e. in(0,∞)×(0,∞), this contradicts the fact that0<R∞

0 t^p−1−λf^p(t)dt <∞. Thus the inequality (2.5) is strict.

Define the weight functionw_λ(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,∞).

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_p Z ∞

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

.

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

Z ∞

0

Z ∞

0

f(x)e eg(y)

max{x^λ, y^λ}dxdy (2.7)

= Z ∞

1

y^λ−q−εq Z ∞

1

1

max{x^λ, y^λ}x^λ−p−εp dx

dy

= Z ∞

1

y^λ−q−εq Z y

1

1

y^λx^λ−p−εp dx

dy+ Z ∞

1

y^λ−q−εq Z ∞

y

1

x^λx^λ−p−εp dx

dy

= 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 number k (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

f(x)e f(y)e

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 ^p_λ +_(p−1)λ^p ≤kor ^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

f(x)

max{x^λ, y^λ}dx p−1

, y ∈(0,∞).

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

f(x) max{x^λ, y^λ}dx

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)

max{x^λ, y^λ}dxdy (2.12)

= Z ∞

0

y^λ+1−qq

Z ∞

0

f(x)

max{x^λ, y^λ}dx h

y^q−1−λq g(y)i dy

≤ Z ∞

0

y^{λ(p−1)−1} Z ∞

0

f(x) max{x^λ, y^λ}

p¹_p Z ∞

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 2.2. Forp=q = 2, (2.3) reduces to (1.5); (2.1), (2.3) are generalizations of (1.5), but (2.1) is not a generalization of (1.3).

Theorem 2.3. If p > 1, 1/p+ 1/q = 1, 0 < λ ≤ min{p, q}, a_n ≥ 0, b_n ≥ 0 such that 0<

∞

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

!¹_{p ∞} 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).

In particular, forλ= 1,(2.13)and(2.14)respectively reduce to the following 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

!¹_{p ∞} 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 coefficientweλ(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:

we_λ(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

=w_λ(r, n) = pq

λ , r =p, q

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, setting ea_n = n^λ−p−εp , eb_n = n^λ−q−εq , n ∈ N. Since 0< λ≤min{p, q}, by (2.9), we get:

∞

X

n=1

∞

X

m=1

eamebn

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

!_p¹ _∞ 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.

Settingbnas:

b_n=n^{λ(p−1)−1}

∞

X

m=1

a_m max{m^λ, n^λ}

!p−1

, 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 Theorem 2.1, 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).

REFERENCES

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

[2] D.S. MITRINOVI ´C, J.E. PE ˘CARI ´CAND A.M. FINK, Inequalities Involving Functions and Their Integrals and Derivatives, Kluwer Academic Publishers, Boston, 1991.

[3] J. KUANG AND L. DEBNATH, On new generalizations of Hilbert’s inequality and their applica- tions, 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. YANGANDL. DEBNATH, On a new strengthened Hardy-Hilbert’s inequality, Internat. J. Math.

Sci., 21(1) (1998), 403–408.