JJ II

volume 4, issue 1, article 16, 2003.

Received 31 October, 2002;

accepted 8 January, 2003.

Communicated by:P.S. Bullen

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Journal of Inequalities in Pure and Applied Mathematics

HILBERT-PACHPATTE TYPE INTEGRAL INEQUALITIES AND THEIR IMPROVEMENT

S.S. DRAGOMIR AND YOUNG-HO KIM

School of Computer Science and Mathematics Victoria University of Technology

PO Box 14428 , Melbourne City MC Victoria 8001, Australia.

EMail:sever.dragomir@vu.edu.au

URL:http://rgmia.vu.edu.au/SSDragomirWeb.html Department of Applied Mathematics

Changwon National University Changwon 641-773, Korea.

EMail:yhkim@sarim.changwon.ac.kr

c

2000Victoria University ISSN (electronic): 1443-5756 114-02

Hilbert-Pachpatte Type Integral Inequalities and their

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Abstract

In this paper, we obtain an extension of multivariable integral inequality of Hilbert-Pachpatte type. By specializing the upper estimate functions in the hy- pothesis and the parameters, we obtain many special cases.

2000 Mathematics Subject Classification:26D15.

Key words: Hilbert’s inequality, Hilbert-Pachpatte type inequality, Hölder’s inequality, Jensen inequality.

The authors would like to thank Professor P.S. Bullen, University of British Columbia, Canada, for the careful reading of the manuscript which led to a considerable im- provement in the presentation of this paper.

Contents

1 Intoduction. . . 3 2 Main Results . . . 5 3 The Various Inequalities. . . 13

References

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1. Intoduction

Hilbert’s double series theorem [3, p. 226] was proved first by Hilbert in his lectures on integral equations. The determination of the constant, the integral analogue, the extension, other proofs of the whole or of parts of the theorems and generalizations in different directions have been given by several authors (cf. [3, Chap. 9]). Specifically, in [10] – [14] the author has established some new inequalities similar to Hilbert’s double-series inequality and its integral analogue which we believe will serve as a model for further investigation. Re- cently, G.D. Handley, J.J. Koliha and J.E. Peˇcari´c [2] established a new class of related integral inequalities from which the results of Pachpatte [12] – [14] are obtained by specializing the parameters and the functions Φ_i.A representative sample is the following.

Theorem 1.1 (Handley, Koliha and Peˇcari´c [2, Theorem 3.1]). Let ui ∈ C^mi([0, x_i])fori∈I.If

u^(k_i ⁱ⁾(s_i) ≤

Z si

0

(s_i−τ_i)^mi−ki−1Φ_i(τ_i)dτ_i, s_i ∈[0, x_i], i∈I,

then

Z x1

0

· · · Z xn

0

Qn i=1

u^(k_i ⁱ⁾(s_i) Pn

i=1ω_is^(α_i ^i+1)/(qiωi) ds₁· · ·ds_n

≤U

n

Y

i=1

x

1 qi

i n

Y

i=1

Z xi

0

(x_i −s_i)^βi+1Φ_i(s_i)^pids_i ¹

pi ,

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whereU = 1 .Qn

i=1[(αi+ 1)

1

qi(βi+ 1)

1 pi].

The purpose of the present paper is to derive an extension of the inequality given in Theorem1.1. In addition, we obtain some new inequalities as Hilbert- Pachpatte type inequalities, these inequalities improve the results obtained by Handley, Koliha and Peˇcari´c [2].

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2. Main Results

In what follows we denote byRthe set of real numbers;R+denotes the interval [0,∞).The symbolsN,Zhave their usual meaning. The following notation and hypotheses will be used throughout the paper:

I ={1, ..., n} n∈N m_i, i∈I m_i ∈N

k_i, i∈I k_i ∈ {0,1, . . . , m_i−1}

x_i, i∈I x_i ∈R, x_i >0

p_i, q_i, i∈I p_i, q_i ∈R, p_i, q_i >0, _p¹

i +_q¹

i = 1

p, q ¹_p =Pn

i=1

1 pi

, ¹_q =Pn i=1

1 qi

a_i, b_i, i∈I a_i, b_i ∈R+, a_i+b_i = 1 ω_i, i∈I ω_i ∈R, ω_i >0, Pn

i=1ω_i = Ω_n α_i, i∈I α_i = (a_i+b_iq_i)(m_i−k_i−1) β_i, i∈I β_i =a_i(m_i−k_i−1)

u_i, i∈I u_i ∈C^m0i([0, x_i]) for some m⁰_i ≥m_i Φ_i, i∈I Φ_i ∈C¹([0, x_i]), Φ_i ≥m_i.

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Here theu_i are given functions of sufficient smoothness, and theΦ_i are subject to choice. The coefficients pi, qi are conjugate Hölder exponents to be used in applications of Hölder’s inequality, and the coefficients a_i, b_i will be used in exponents to factorize integrands. The coefficients ω_i will act as weights in applications of the geometric-arithmetic mean inequality. The coefficients αi

andβ_i arise naturally in the derivation of the inequalities. Our main results are given in the following theorems.

Theorem 2.1. Letu_i ∈C^mi([0, x_i])fori∈I.If

(2.1)

u^(k_i ⁱ⁾(s_i) ≤

Z si

0

(s_i−τ_i)^mi−ki−1Φ_i(τ_i)dτ_i, s_i ∈[0, x_i], i∈I, then

(2.2) Z x1

0

· · · Z xn

0

Qn i=1

u^(k_i ⁱ⁾(s_i) h 1

Ωn

Pn

i=1ω_is^(α_i ^i+1)/(qiωi)iΩn ds₁· · ·ds_n

≤V

n

Y

i=1

x

1 qi

i n

Y

i=1

Z xi

0

(x_i −s_i)^βi+1Φ_i(s_i)^pids_{i pi}¹

,

where

(2.3) V = 1

Qn i=1

h

(α_i+ 1)^qi1(β_i+ 1)^pi1i. Proof. Factorize the integrand on the right side of (2.1) as

(si−τi)^{(ai/qi+bi)(mi−ki−1)}×(si−τi)^{(ai/pi)(mi−ki−1)}Φi(τi)

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and apply Hölder’s inequality [9, p.106]. Then

u^(k_i ⁱ⁾(si) ≤

Z si

0

(si−τi)^{(ai+biqi)(mi−ki−1)}dτi

_qi¹

× Z si

0

(si−τi)^{ai(mi−ki−1)}Φi(τi)^pidτi

_pi¹

= s^(α_i ^i+1)/qi (αi+ 1)^qi1

Z si

0

(s_i−τ_i)^βiΦ_i(τ_i)^pidτ_{i pi}¹

.

Using the inequality of means [9, p. 15]

n

Y

i=1

s^w_iⁱ

!_Ωn¹

≤ 1

Ωn n

X

i=1

w_is^r_i

!¹_r

forr >0,we deduce that

n

Y

i=1

s^w_i^ir≤

"

1 Ω_n

n

X

i=1

w_is^r_i

#Ωn

forr >0.According to above inequality, we have

n

Y

i=1

u^(k_i ⁱ⁾(s_i)

≤ 1 Qn

i=1(α_i+ 1)^qi1

"

1 Ω_n

n

X

i=1

ω_is^(α_i ^i+1)/(qiωi)

#Ωn

×

n

Y

i=1

Z si

0

(s_i −τ_i)^βiΦ_i(τ_i)^pidτ_i ¹

pi

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for r = (α_i + 1)/q_iω_i. In the following estimate we apply Hölder’s inequality and, at the end, change the order of integration:

Z x1

0

· · · Z xn

0

Qn i=1

u^(k_i ⁱ⁾s_i

h 1

Ωn

Pn

i=1ω_is^(α_i ^i+1)/(qiωi)iΩn ds1· · ·dsn

≤ 1

Qn

i=1(α_i+ 1)^qi1

n

Y

i=1

"

Z xi

0

Z si

0

(s_i−τ_i)^βiΦ_i(τ_i)^pidτ_i, _pi¹

ds_i

#

≤ 1

Qn

i=1(α_i+ 1)^qi1

n

Y

i=1

x

1 qi

i

Z xi

0

Z si

0

(si−τi)^βiΦi(τi)^pidτi,

dsi

_pi¹

= 1

Qn

i=1[(α_i+ 1)^qi1(β_i+ 1)^pi1]

n

Y

i=1

x

1 qi

i n

Y

i=1

Z xi

0

(x_i−s_i)^βi+1Φ_i(s_i)^pids_i ¹

pi .

This proves the theorem.

Remark 2.1. In Theorem2.1, settingΩ_n = 1, we have Theorem1.1.

Corollary 2.2. Under the assumptions of Theorem2.1, ifr >0, we have

Z x1

0

· · · Z xn

0

Qn i=1

u^(k_i ⁱ⁾s_i

h 1

Ωn

Pn

i=1ω_is^(α_i ^i+1)/(qiωi)iΩn ds₁· · ·ds_n

≤p^r·p1 V

n

Y

i=1

x

1 qi

i

" _n X

i=1

1 p_i

Z xi

0

(x_i−s_i)^βi+1Φ_is^p_iⁱds_i r#_r·p¹

,

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whereV is defined by (2.3).

Proof. By the inequality of means, for anyA_i ≥0andr >0,we obtain

n

Y

i=1

A

1 pi

i ≤

"

p

n

X

i=1

1 p_iA^r_i

#_r·p¹ .

The corollary then follows from the preceding theorem.

Lemma 2.3. Let γ₁ > 0 and γ₂ < −1. Let ω_i > 0, Pn

i=1ω_i = Ω_n and let s_i >0, i= 1, . . . , nbe real numbers. Then

n

Y

i=1

s^ω_i^iγ1γ2 ≥

"

1 Ω_n

n

X

i=1

ω_is^−γ_i ²

#−γ1Ωn

.

Proof. By the inequality of means, for anyγ₁ >0andγ₂ <−1,we have

n

Y

i=1

s^ω_i^iγ1γ2 ≥

"

1 Ω_n

n

X

i=1

ω_is_i

#γ1γ2Ωn

.

Using the fact that x^−γ12 is concave and using the Jensen inequality, we have that

"

1 Ω_n

n

X

i=1

ω_is_i

#γ1γ2Ωn

=

"

1 Ω_n

n

X

i=1

ω_if(s^−γ_i ²)

#γ1γ2Ωn

≥

"

f 1

Ω_n

n

X

i=1

ω_is^−γ_i ²

!#γ1γ2Ωn

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=



 1 Ω_n

n

X

i=1

ω_is^−γ_i ²

!−_γ¹

2





γ1γ2Ωn

=

"

1 Ωn

n

X

i=1

ω_is^−γ_i ²

#−γ₁Ωn

.

The proof of the lemma is complete.

Theorem 2.4. Under the assumptions of Theorem2.1, ifγ₂ <−1,then

Z x1

0

· · · Z xn

0

Qn i=1

u^(k_i ⁱ⁾(s_i) h 1

Ωn

Pn

i=1ω_is^−γ_i ²i−(αi+1)Ωn/γ2qiωi ds₁· · ·ds_n

≤V

n

Y

i=1

x

1 qi

i n

Y

i=1

Z xi

0

(xi−si)^βi+1Φi(si)^pidsi

_pi¹ ,

whereV is given by (2.3).

Proof. Using the inequality of Lemma2.3, for anyγ₁ >0andγ₂ <−1,we get

n

Y

i=1

s^ω_i^iγ1 ≤

"

1 Ωn

n

X

i=1

ω_is^−γ_i ²

#−^γ1Ωn

γ2

.

According to above inequality, we deduce that

n

Y

i=1

u^(k_i ⁱ⁾(si)

≤ 1 Qn

i=1(α_i+ 1)^qi1

"

1 Ω_n

n

X

i=1

ωis^−γ_i ²

#^−W1

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×

n

Y

i=1

"

Z (si) 0

(s_i−τ_i)^βiΦ_i(τ_i)^pidτ_i

#¹

pi

,

where W₁ = (α_i+ 1)Ω_n/γ₂q_iω_i. The proof of the theorem then follows from the preceding Theorem2.1.

Corollary 2.5. Under the assumptions of Theorem2.4, ifr >0, we have

Z x1

0

· · · Z xn

0

Qn i=1

u^(k_i ⁱ⁾(s_i) h 1

Ωn

Pn

i=1ω_is^−γ_i ²i^−(αi+1)Ωn/γ2qiωi ds₁· · ·ds_n

≤p^r·p1 V

n

Y

i=1

x

1 qi

i

" _n X

i=1

1 p_i

Z xi

0

(x_i−s_i)^βi+1Φ_i(s_i)^pids_i r#_r·p¹

,

whereV is given by (2.3).

Proof. By the inequality of means, for anyA_i ≥0andr >0,we obtain

n

Y

i=1

A

1 pi

i ≤

"

p

n

X

i=1

1 p_iA^r_i

#_r·p¹ .

The corollary then follows from the preceding Theorem2.4.

In the following section we discuss some choice of the functionsΦ_i.

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3. The Various Inequalities

Theorem 3.1. Letu_i ∈C^mi([0, x_i])be such thatu^(j)_i (0) = 0forj ∈ {0, . . . , m_i− 1}, i∈I.Then

(3.1) Z x1

0

· · · Z xn

0

Qn i=1

u^(k_i ⁱ⁾(s_i) h 1

Ωn

Pn

i=1ωis^(α_i ^i+1)/(qiωi)

iΩn ds1· · ·dsn

≤V₁

n

Y

i=1

x

1 qi

i n

Y

i=1

Z xi

0

(x_i−s_i)^βi+1

u^(m_i ⁱ⁾(s_i)

pi

ds_{i pi}¹

,

where

(3.2) V1 = 1

Qn i=1

h

(m_i−k_i−1)!(α_i+ 1)^qi1(β_i + 1)^pi1 i.

Proof. Inequality (3.1) is proved when we set

Φ_i(s_i) =

u^(m_i ⁱ⁾(s_i) (m_i−k_i−1)!

in Theorem2.1.

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Corollary 3.2. Under the assumptions of Theorem3.1, ifr >0, we have

Z x1

0

· · · Z xn

0

Qn i=1

u^(k_i ⁱ⁾(s_i) h 1

Ωn

Pn

i=1ω_is^(α_i ^i+1)/(qiωi)iΩn ds₁· · ·ds_n

≤p^r·p1 V₁

n

Y

i=1

x

1 qi

i

" _n X

i=1

1 p_i

Z xi

0

(x_i−s_i)^βi+1

u^(m_i ⁱ⁾(s_i)

pi

ds_i r#_r·p¹

,

whereV₁ is given by (3.2).

Theorem 3.3. Under the assumptions of Theorem3.1, ifγ₂ <−1,then

(3.3) Z x1

0

· · · Z xn

0

Qn i=1

u^(k_i ⁱ⁾(s_i) h 1

Ωn

Pn

i=1ω_is^−γ_i ²i−(α_i+1)Ωn/γ2qiωi ds1· · ·dsn

≤V₁

n

Y

i=1

x

1 qi

i n

Y

i=1

Z xi

0

(x_i−s_i)^βi+1

u^(m_i ⁱ⁾(s_i)

pi

ds_i ¹

pi ,

whereV1 is given by (3.2).

Proof. Inequality (3.3) is proved when we set

Φi(si) =

u^(m_i ⁱ⁾(s_i) (m_i−k_i−1)!

in Theorem2.4.

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Corollary 3.4. Under the assumptions of Theorem3.3, ifr >0, we have

Z x1

0

· · · Z xn

0

Qn i=1

u^(k_i ⁱ⁾(s_i) h 1

Ωn

Pn

i=1ωis^−γ_i ²

i−(α_i+1)Ωn/γ2qiωi ds₁· · ·ds_n

≤p^r·p1 V1 n

Y

i=1

x

1 qi

i

" _n X

i=1

1 p_i

Z xi

0

(xi−si)^βi+1

u^(m_i ⁱ⁾(si)

pi

dsi

r#_r·p¹ .

We discuss a number of special cases of Theorem 3.1. Similar examples apply also to Corollary3.2, Theorem3.3and Corollary3.4.

Example 3.1. Ifa_i = 0andb_i = 1fori∈I,then Theorem3.1becomes

Z x1

0

· · · Z xn

0

Qn i=1

u^(k_i ⁱ⁾(s_i) h 1

Ωn

Pn

i=1ω_is^(q_i ^{imi−qiki−qi+1)/(qiωi)}iΩn ds₁· · ·ds_n

≤V₂

n

Y

i=1

x

1 qi

i n

Y

i=1

Z xi

0

(x_i−s_i)

u^(m_i ⁱ⁾(s_i)

pi

ds_i ¹

pi ,

where

V2 = 1

Qn i=1

h

(m_i−k_i−1)!(q_im_i−q_ik_i−q_i+ 1)^qi1i.

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Example 3.2. If a_i = 0, b_i = 1, q_i =n, p_i = n/(n−1), m_i =mandk_i = k fori∈I,then

Z x1

0

· · · Z xn

0

Qn i=1

u^(k_i ⁱ⁾(si) h 1

Ωn

Pn

i=1ω_is(nm−nk−n+1)/(nωi) i

iΩn ds₁· · ·ds_n

≤

√n

x₁· · ·x_n (m−k−1)!n

(nm−nk−n+ 1)

×

n

Y

i=1

Z xi

0

(xi−si)

u^(m)_i (si)

n n−1 dsi

ⁿ⁻¹_n .

For q = p = n = 2 andω_i = _n¹ this is [12, Theorem 1]. Settingq = p = 2, k = 0, n= 1andω_i = _n¹, we recover the result of [14].

Example 3.3. Ifa_i = 0andb_i = 1fori∈I,then Theorem3.1becomes

Z x1

0

· · · Z xn

0

Qn i=1

u^(k_i ⁱ⁾(s_i) h 1

Ωn

Pn

i=1ω_is^(m_i ^{i−ki)/(qiωi)}iΩn ds₁· · ·ds_n

≤V₃

n

Y

i=1

x

1 qi

i n

Y

i=1

Z xi

0

(x_i−s_i)^mi−ki

u^(m_i ⁱ⁾(s_i)

pi

ds_{i pi}¹

,

where

V₃ = 1

Qn i=1

(m_i−k_i)!.

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Example 3.4. If a_i = 1, b_i = 0, q_i =n, p_i = n/(n−1), m_i =mandk_i = k fori∈I.Then (3.1) becomes

Z x1

0

· · · Z xn

0

Qn i=1

u^(k_i ⁱ⁾(s_i) h 1

Ωn

Pn

i=1ω_is^(m−k)/(nω_i ⁱ⁾iΩn ds₁· · ·ds_n

≤

√n

x1· · ·xn

(m−k)!n n

Y

i=1

Z xi

0

(x_i−s_i)^m−k

u^(m)_i (s_i)

n/(n−1)

ds_i ⁽ⁿ⁻¹⁾_n

.

Example 3.5. Letp₁, p₂ ∈R+.If we setn = 2, ω₁ = _p¹

1, ω₂ = _p¹

2, m_i = 1and k_i = 0fori = 1,2in Theorem3.1, then by our assumptionsq₁ =p₁/(p₁−1), q₂ =p₂/(p₂−1),and we obtain

Z x1

0

Z x2

0

|u₁(s₁)| |u₂(s₂)|

h 1 p1p2Ω2

p₂s^(p₁¹⁻¹⁾+p₁s^(p₂²⁻¹⁾iΩ2 ds₁ds₂

≤x^(p₁^1−1)/p1x^(p₂ ^2−1)/p2 Z x1

0

(x1−s1)|u⁰₁(s1)|^p1 ds1

_p¹

1

× Z x2

0

(x2−s2)|u⁰₂(s2)|^p2 ds2

_p¹

2

. If we set ω₁ +ω₂ = 1 in Example 3.5, then we have [13, Theorem 2]. (The values ofa_i andb_iare irrelevant.)

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References

[1] BICHENG YANG, On Hilbert’s integral inequality, J. Math. Anal. Appl., 220 (1988), 778–785.

[2] G.D. HANDLEY, J.J. KOLIHA AND PE ˇCARI ´C, New Hilbert-Pachpatte type integral inequalities, J. Math. Anal. Appl., 257 (2001), 238–250.

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

[4] YOUNG-HO KIM, Refinements and Extensions of an inequality, J. Math.

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[5] V. LEVIN, On the two-parameter extension and analogue of Hilbert’s in- equality, J. London Math. Soc., 11 (1936), 119–124.

[6] G. MINGZE, On Hilbert’s inequality and its applications, J. Math. Anal.

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[7] D.S. MITRINOVI ´C, Analytic inequalities, Springer-Verlag, Berlin, New York, 1970.

[8] D.S. MITRINOVI ´C AND J.E. PE ˇCARI ´C, On inequalities of Hilbert and Widder, Proc. Edinburgh Math. Soc., 34 (1991), 411–414.

[9] D.S. MITRINOVI ´C, J.E. PE ˇCARI ´CANDA.M. FINK, Classical and New Inequalities in Analysis, Kluwer Academic, Dordrecht, 1993.

[10] B.G. PACHPATTE, A note on Hilbert type inequality, Tamkang J. Math., 29 (1998), 293–298.

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Improvement S.S. Dragomir and Young-Ho Kim

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

[12] B.G. PACHPATTE, Inequalities similar to the integral analogue of Hilbert’s Inequality, Tamkang J. Math., 30 (1999), 139–146.

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

[14] B.G. PACHPATTE, A note on inequality of Hilbert type, Demonstratio Math., in press.

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