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volume 4, issue 2, article 33, 2003.

Received 25 May, 2002;

accepted 7 April, 2003.

Communicated by:S.S. Dragomir

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Journal of Inequalities in Pure and Applied Mathematics

SOME NEW INEQUALITIES SIMILAR TO HILBERT-PACHPATTE TYPE INEQUALITIES

ZHONGXUE LÜ

Department of Basic Science of Technology College, Xuzhou Normal University, 221011,

People’s Republic of China.

E-Mail:[email protected]

2000c Victoria University ISSN (electronic): 1443-5756 059-02

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Some New Inequalities Similar to Hilbert-Pachpatte Type

Inequalities Zhongxue Lü

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J. Ineq. Pure and Appl. Math. 4(2) Art. 33, 2003

http://jipam.vu.edu.au

Abstract

In this paper, some new inequalities similar to Hilbert-Pachpatte type inequalities are given.

2000 Mathematics Subject Classification:26D15.

Key words: Inequalities, Hilbert-Pachpatte inequalities, Hölder inequality.

1. Introduction

In [1, Chap. 9], the well-known Hardy-Hilbert inequality is given as follows.

Theorem 1.1. Let p >1, q >1, ¹_p + ¹_q = 1, a_m, b_n≥0, 0 < P∞

n=1a^p_n<∞, 0<P∞

n=1b^q_n<∞.Then

(1.1)

∞

X

m=1

∞

X

n=1

ambn

(m+n)^λ ≤ π sin(π/p)

∞

X

m=1

a^p_m

!¹_p _∞ X

n=1

b^q_n

!¹_q

where _sin(π/p)^π is best possible.

The integral analogue of the Hardy-Hilbert inequality can be stated as follows

Theorem 1.2. Letp >1, q >1, ¹_p + ¹_q= 1, f(x), g(y)≥0, 0<R∞

0 f^p(x)dx <

∞, 0<R∞

0 g^q(y)dy <∞.Then (1.2)

Z ∞

0

Z ∞

0

f(x)g(y)

x+y dxdy≤ π sin(π/p)

Z ∞

0

f^p(x)dx

¹_pZ ∞

0

g^q(y)dy ¹_q

,

where _sin(π/p)^π is best possible.

In [1, Chap. 9] the following extension of Hardy-Hilbert’s double-series theorem is given.

Theorem 1.3. Letp >1, q >1, ¹_p +¹_q ≥1, 0< λ= 2−¹_p− ¹_q = ¹_p +¹_q ≤1.

Then

∞

X

m=1

∞

X

n=1

a_mb_n

(m+n)^λ ≤K

∞

X

m=1

a^p_m

!_p¹ _∞ X

n=1

b^q_n

!¹_q ,

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whereK =K(p, q)depends onpandqonly.

The following integral analogue of Theorem 1.3 is also given in [1, Chap.

9].

Theorem 1.4. Under the same conditions as in Theorem1.1we have Z ∞

0

Z ∞

0

f(x)g(y)

(x+y)^λdxdy ≤K Z ∞

0

f^pdx

¹_pZ ∞

0

g^qdy ¹_q

,

whereK =K(p, q)depends onpandqonly.

The inequalities in Theorems1.1 and1.2were studied by Yang and Kuang (see [2,3]). In [4,5], some new inequalities similar to the inequalities given in Theorems1.1,1.2,1.3and1.4were established.

In this paper, we establish some new inequalities similar to the Hilbert- Pachpatte inequality.

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2. Main Results

In what follows we denote by Rthe set of real numbers. Let N = {1,2, . . .}, N⁰ ={0,1,2, . . .}. We define the operator∇by∇u(t) = u(t)−u(t−1)for any functionudefined on N. For any functionu(t) : [0,∞) → R, we denote byu⁰the derivatives ofu.

First we introduce some Lemmas.

Lemma 2.1. (see [2]). Letp > 1, q >1, ¹_p +¹_q = 1,λ >2−min{p, q}, define the weight functionω₁(q, x)as

ω1(q, x) :=

Z ∞

0

1 (x+y)^λ

x y

^2−λ_q

dy, x∈[0,∞).

Then

(2.1) ω₁(q, x) =B

q+λ−2

q ,p+λ−2 p

x^1−λ,

whereB(p, q)isβ-function.

Lemma 2.2. (see [3]). Letp > 1, q >1, ¹_p +¹_q = 1,λ >2−min{p, q}, define the weight functionω₂(q, x)as

ω₂(q, x) :=

Z ∞

0

1 x^λ +y^λ

x y

^2−λ_q

dy, x∈[0,∞).

Then

(2.2) ω2(q, x) = 1 λB

q+λ−2

qλ ,p+λ−2 pλ

x^1−λ.

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Lemma 2.3. Let p > 1, q > 1, ¹_p + ¹_q = 1, λ > 2−min{p, q}, define the weight functionω₃(q, m)as

ω3(q, m) :=

∞

X

n=1

1 (m+n)^λ

m n

^2−λ_q

, m∈ {1,2, . . .}.

Then

(2.3) ω₃(q, m)< B

q+λ−2

q ,p+λ−2 p

m^1−λ,

whereB(p, q)isβ-function.

Proof. By Lemma2.1,we have

ω₃(q, m)<

Z ∞

0

1 (m+y)^λ

m y

^2−λ_q dy

=B

q+λ−2

q ,p+λ−2 p

m^1−λ.

The proof is completed.

Lemma 2.4. Let p > 1, q > 1, ¹_p + ¹_q = 1, λ > 2−min{p, q}, define the weight functionω₄(q, m)as

ω₄(q, m) :=

∞

X

n=1

1 m^λ +n^λ

m n

^2−λ_q

, m∈ {1,2, . . .}.

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Then

(2.4) ω₄(q, m)< 1 λB

q+λ−2

qλ ,p+λ−2 pλ

m^1−λ. Proof. By Lemma2.2, we have

ω₄(q, m)<

Z ∞

0

1 m^λ+y^λ

m y

^2−λ_q dy

= 1 λB

q+λ−2

qλ ,p+λ−2 pλ

m^1−λ.

The proof is completed.

Our main result is given in the following theorem.

Theorem 2.5. Letp >1, ¹_p +¹_q = 1, andf(x), g(y)be real-valued continuous functions defined on[0,∞), respectively, and letf(0) =g(0) = 0, and

0<

Z ∞

0

Z x

0

|f⁰(τ)|^pdτ dx <∞, 0<

Z ∞

0

Z y

0

|g⁰(δ)|^qdδdy <∞.

Then

(2.5) Z ∞

0

Z ∞

0

|f(x)| |g(y)|

(qx^p−1+py^q−1)(x+y)dxdy

≤ π

sin(π/p)pq Z ∞

0

Z x

0

|f⁰(τ)|^pdτ dx

¹_p Z ∞

0

Z y

0

|g⁰(δ)|^qdδdy ¹_q

.

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In particular, whenp=q = 2, we have

(2.6) Z ∞

0

Z ∞

0

|f(x)| |g(y)|

(x+y)² dxdy

≤ π 2

Z ∞

0

Z x

0

|f⁰(τ)|²dτ dx

¹₂ Z ∞

0

Z y

0

|g⁰(δ)|²dδdy ¹₂

.

Proof. From the hypotheses, we have the following identities

(2.7) f(x) =

Z x

0

f⁰(τ)dτ,

and

(2.8) g(y) =

Z y

0

g⁰(δ)dδ

forx, y ∈ (0,∞). From (2.7) and (2.8) and using Hölder’s integral inequality, respectively, we have

(2.9) |f(x)| ≤x¹^q

Z x

0

|f⁰(τ)|^pdτ ¹_p

and

(2.10) |g(y)| ≤y¹^p

Z y

0

|g⁰(δ)|^qdδ ¹_q

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forx, y ∈(0,∞). From (2.9) and (2.10) and using the elementary inequality (2.11) z1z2 ≤ z₁^p

p +z₂^q

q , z1 ≥0, z2 ≥0, 1 p +1

q = 1, p > 1,

we observe that

|f(x)| |g(y)| ≤x¹^qy¹^p Z x

0

|f⁰(τ)|^pdτ

¹_pZ y

0

≤

x^p−1

p + y^q−1 q

Z x

0

|f⁰(τ)|^pdτ

¹_pZ y

0

|g⁰(δ)|^qdδ ¹_q (2.12)

forx, y ∈(0,∞). From (2.12) we observe that (2.13) |f(x)| |g(y)|

qx^p−1+py^q−1 ≤ 1 pq

Z x

0

|f⁰(τ)|^pdτ

¹_pZ y

0

.

Hence (2.14)

Z ∞

0

Z ∞

0

|f(x)| |g(y)|

(qx^p−1+py^q−1)(x+y)dxdy

≤ 1 pq

Z ∞

0

Z ∞

0

Rx

0 |f⁰(τ)|^pdτ¹_p Ry

0 |g⁰(δ)|^qdδ¹_q

x+y dxdy.

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By Hölder’s integral inequality and (2.1), we have Z ∞

0

Z ∞

0

Rx

0 |f⁰(τ)|^pdτ¹_p Ry

0 |g⁰(δ)|^qdδ¹_q

x+y dxdy

= Z ∞

0

Z ∞

0

Rx

0 |f⁰(τ)|^pdτ¹_p (x+y)^p¹

x y

_pq¹ Ry

0 |g⁰(δ)|^qdδ¹_q (x+y)¹^q

y x

_pq¹ dxdy

≤ Z ∞

0

Z ∞

0

Rx

0 |f⁰(τ)|^pdτ x+y

x y

¹_q dxdy

!_p¹

× Z ∞

0

Z ∞

0

Ry

0 |g⁰(δ)|^qdδ x+y

y x

¹_p dxdy

¹_q

≤ π

sin(π/p) Z ∞

0

Z x

0

|f⁰(τ)|^pdτ dx

_p¹ Z ∞

0

Z y

0

|g⁰(δ)|^qdδdy ¹_q (2.15)

by (2.14) and (2.15), we get (2.5). The proof of Theorem2.5is complete.

In a similar way to the proof of Theorem 2.5, we can prove the following theorems.

Theorem 2.6. Letp > 1, ¹_p + ¹_q = 1, λ > 2−min{p, q}, andf(x), g(y)be real-valued continuous functions defined on[0,∞),respectively, and letf(0) = g(0) = 0, and

0<

Z ∞

0

Z x

0

x^1−λ|f⁰(τ)|^pdτ dx <∞, 0<

Z ∞

0

Z y

0

y^1−λ|g⁰(δ)|^qdδdy <∞,

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then

(2.16) Z ∞

0

Z ∞

0

|f(x)| |g(y)|

(qx^p−1+py^q−1)(x+y)^λdxdy

≤ B

q+λ−2

q ,^p+λ−2_p pq

Z ∞

0

Z x

0

x^1−λ|f⁰(τ)|^pdτ dx ¹_p

× Z ∞

0

Z y

0

y^1−λ|g⁰(δ)|^qdδdy ¹_q

. In particular, whenp=q = 2,

(2.17) Z ∞

0

Z ∞

0

|f(x)| |g(y)|

(x+y)^1+λ dxdy

≤ B ^λ₂,^λ₂ 2

Z ∞

0

Z x

0

x^1−λ|f⁰(τ)|²dτ dx ¹₂

× Z ∞

0

Z y

0

y^1−λ|g⁰(δ)|²dδdy ¹₂

.

Theorem 2.7. Letp > 1, ¹_p + ¹_q = 1, λ > 2−min{p, q}, andf(x), g(y)be real-valued continuous functions defined on[0,∞), respectively, and letf(0) = g(0) = 0, and

0<

Z ∞

0

Z x

0

x^1−λ|f⁰(τ)|^pdτ dx <∞, 0<

Z ∞

0

Z y

0

y^1−λ|g⁰(δ)|^qdδdy <∞.

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Then

(2.18) Z ∞

0

Z ∞

0

|f(x)| |g(y)|

(qx^p−1+py^q−1)(x^λ+y^λ)dxdy

≤ B

q+λ−2

qλ ,^p+λ−2_pλ λpq

Z ∞

0

Z x

0

x^1−λ|f⁰(τ)|^pdτ dx ¹_p

× Z ∞

0

Z y

0

y^1−λ|g⁰(δ)|^qdδdy ¹_q

. In particular, whenp=q = 2,

(2.19) Z ∞

0

Z ∞

0

|f(x)| |g(y)|

(x^λ+y^λ)(x+y)dxdy

≤ π 2λ

Z ∞

0

Z x

0

x^1−λ|f⁰(τ)|²dτ dx ¹₂

× Z ∞

0

Z y

0

y^1−λ|g⁰(δ)|²dδdy ¹₂

.

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3. Discrete Analogues

Theorem 3.1. Let p > 1, ¹_p + ¹_q = 1, and {a(m)} and {b(n)} be two se- quences of real numbers where m, n ∈ N0, and a(0) = b(0) = 0, and 0 <

P∞ m=1

Pm

τ=1|∇a(τ)|^p <∞,0<P∞ n=1

Pn

δ=1|∇b(δ)|^q <∞, then (3.1)

∞

X

m=1

∞

X

n=1

|a_m| |b_n|

(qm^p−1+pn^q−1)(m+n)

≤ π

sin(π/p)pq

∞

X

m=1 m

X

k=1

a^p_k

!¹_p _∞ X

n=1 n

X

r=1

b^q_r

!¹_q .

(3.2)

∞

X

m=1

∞

X

n=1

|a_m| |b_n| (m+n)² ≤ π

2

∞

X

m=1 m

X

k=1

a²_k

!¹₂ _∞ X

n=1 n

X

r=1

b²_r

!¹₂ .

Proof. From the hypotheses, it is easy to observe that the following identities hold

(3.3) a_m =

m

X

τ=1

∇a(τ),

and

(3.4) b_n=

n

X

δ=1

∇b(δ)

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form, n∈N. From (3.3) and (3.4) and using Hölder’s inequality, we have

(3.5) |am| ≤m¹^q

m

X

τ=1

|∇a(τ)|^p

!¹_p ,

and

(3.6) |b_n| ≤n¹^p

n

X

δ=1

|∇b(δ)|^q

!¹_q

form, n∈N. From (3.5) and (3.6) and using the elementary inequality (2.11), we observe that

|a_m| |b_n| ≤m¹^qn^p¹

m

X

τ=1

|∇a(τ)|^p

!¹_p _n X

δ=1

|∇b(δ)|^q

!¹_q

≤

m^p−1

p + n^q−1 q

m

X

τ=1

|∇a(τ)|^p

!¹_p _n X

δ=1

|∇b(δ)|^q

!¹_q (3.7)

form, n∈N. From (3.7), we observe that (3.8) |a_m| |b_n|

qm^p−1+pn^q−1 ≤ 1 pq

m

X

τ=1

|∇a(τ)|^p

!¹_p _n X

δ=1

|∇b(δ)|^q

!¹_q .

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Hence (3.9)

∞

X

m=1

∞

X

n=1

|a_m| |b_n|

(qm^p−1+pn^q−1)(m+n)

≤ 1 pq

∞

X

m=1

∞

X

n=1

(Pm

τ=1|∇a(τ)|^p)¹^p(Pn

δ=1|∇b(δ)|^q)¹^q

m+n .

By the Hölder inequality and (2.3)

∞

X

m=1

∞

X

n=1

(Pm

τ=1|∇a(τ)|^p)¹^p(Pn

δ=1|∇b(δ)|^q)¹^q m+n

=

∞

X

m=1

∞

X

n=1

(Pm

τ=1|∇a(τ)|^p)¹^p (m+n)¹^p

m n

_pq¹ (Pn

δ=1|∇b(δ)|^q)¹^q (m+n)¹^q

n m

_pq¹

≤

∞

X

m=1

∞

X

n=1

Pm

τ=1|∇a(τ)|^p m+n

m n

¹_q

!¹_p

×

∞

X

m=1

∞

X

n=1

Pn

δ=1|∇b(δ)|^q m+n

n m

¹_p

!¹_q

< π sin(π/p)

∞

X

m=1 m

X

τ=1

|∇a(τ)|^p

!¹_p _∞ X

n=1 n

X

δ=1

|∇b(δ)|^q

!¹_q (3.10)

by (3.9) and (3.10), we get (3.1). The proof of Theorem3.1is complete.

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In a similar manner to the proof of Theorem3.1, we can prove the following theorems.

Theorem 3.2. Letp > 1, ¹_p+¹_q = 1,λ >2−min{p, q}, and{a(m)}and{b(n)}

be two sequences of real numbers wherem, n∈N⁰,anda(0) =b(0) = 0, and

0<

∞

X

m=1 m

X

τ=1

m^1−λ|∇a(τ)|^p <∞,

0<

∞

X

n=1 n

X

δ=1

n^1−λ|∇b(δ)|^q <∞,

then

(3.11)

∞

X

m=1

∞

X

n=1

|am| |bn|

(qm^p−1+pn^q−1)(m+n)^λ

≤

B_q+λ−2

q ,^p+λ−2_p pq

∞

X

m=1 m

X

τ=1

m^1−λ|∇a(τ)|^p

!¹_p

×

∞

X

n=1 n

X

δ=1

n^1−λ|∇b(δ)|^q

!¹_q .

(3.12)

∞

X

m=1

∞

X

n=1

|a_m| |b_n| (m+n)^1+λ

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≤ B ^λ₂,^λ₂ 2

∞

X

m=1 m

X

τ=1

m^1−λ|∇a(τ)|²

!¹₂ _∞ X

n=1 n

X

δ=1

n^1−λ|∇b(δ)|²

!¹₂ .

Theorem 3.3. Letp > 1, ¹_p+¹_q = 1,λ >2−min{p, q}, and{a(m)}and{b(n)}

be two sequences of real numbers wherem, n∈N0, anda(0) = b(0) = 0, and

0<

∞

X

m=1 m

X

τ=1

m^1−λ|∇a(τ)|^p <∞,

0<

∞

X

n=1 n

X

δ=1

n^1−λ|∇b(δ)|^q <∞,

then

(3.13)

∞

X

m=1

∞

X

n=1

|a_m| |b_n|

(qm^p−1+pn^q−1)(m^λ+n^λ)

≤ B

q+λ−2

qλ ,^p+λ−2_pλ λpq

∞

X

m=1 m

X

τ=1

m^1−λ|∇a(τ)|^p

!¹_p

×

∞

X

n=1 n

X

δ=1

n^1−λ|∇b(δ)|^q

!¹_q .

(3.14)

∞

X

m=1

∞

X

n=1

|a_m| |b_n| (m+n)(m^λ+n^λ)

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≤ π 2λ

∞

X

m=1 m

X

τ=1

m^1−λ|∇a(τ)|²

!¹₂ _∞ X

n=1 n

X

δ=1

n^1−λ|∇b(δ)|²

!¹₂ .

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References

[1] G.H. HARDY, J.E. LITTLEWOOD AND G. PÓLYA, Inequalities, Cam- bridge Univ. Press, London, 1952.

[2] BICHENG YANG, A generalized Hilbert’s integral inequality with the best constant, Ann. of Math.(Chinese), 21A:4 (2000), 401–408.

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

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

[4] B.G. PACHPATTE, Inequalities similar to certain extensions of Hilbert’s inequality, J.Math. Anal. Appl., 243 (2000), 217–227.

[5] B.G. PACHPATTE, On some new inequalities similar to Hilbert’s inequal- ity, J. Math. Anal. Appl., 226 (1998), 166–179.

[6] D.S. MITRINOVI ´C, Analytic Inequalities, Springer-Verlag, Berlin/New York,1970.

[7] E.F. BECKENBACH AND R. BELLMAN, Inequalities, Springer-Verlag, Berlin, 1983.