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

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

Volume 4, Issue 1, Article 16, 2003

HILBERT-PACHPATTE TYPE INTEGRAL INEQUALITIES AND THEIR IMPROVEMENT

S.S. DRAGOMIR AND YOUNG-HO KIM SCHOOL OFCOMPUTERSCIENCE ANDMATHEMATICS

VICTORIAUNIVERSITY OFTECHNOLOGY

PO BOX14428 , MELBOURNECITYMC VICTORIA8001, AUSTRALIA. sever.dragomir@vu.edu.au

URL:http://rgmia.vu.edu.au/SSDragomirWeb.html DEPARTMENT OFAPPLIEDMATHEMATICS

CHANGWONNATIONALUNIVERSITY

CHANGWON641-773, KOREA. yhkim@sarim.changwon.ac.kr

Received 31 October, 2002; accepted 8 January, 2003 Communicated by P.S. Bullen

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

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

2000 Mathematics Subject Classification. 26D15.

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. Recently, 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 functionsΦ_i.A representative sample is the following.

ISSN (electronic): 1443-5756 c

2003 Victoria University. All rights reserved.

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 improvement in the presentation of this paper.

114-02

Theorem 1.1 (Handley, Koliha and Peˇcari´c [2, Theorem 3.1]). Letu_i ∈ 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 pi}¹

,

whereU = 1. Qn

i=1[(α_i + 1)^qi1(β_i+ 1)^pi1 ].

The purpose of the present paper is to derive an extension of the inequality given in Theorem 1.1. 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].

2. MAINRESULTS

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

I ={1, ..., n} n∈N mi, i∈I mi ∈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) βi, i∈I βi =ai(mi−ki−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.

Here the u_i are given functions of sufficient smoothness, and the Φ_i are subject to choice.

The coefficients p_i, q_i 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 coefficientsω_i will act as weights in applications of the geometric-arithmetic mean inequality.

The coefficientsα_i 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

(αi+ 1)

1

qi(βi+ 1)

1 pi

i.

Proof. Factorize the integrand on the right side of (2.1) as

(s_i−τ_i)^{(ai/qi+bi)(mi−ki−1)}×(s_i−τ_i)^{(ai/pi)(mi−ki−1)}Φ_i(τ_i) 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

(s_i−τ_i)^{ai(mi−ki−1)}Φ_i(τ_i)^pidτ_{i pi}¹

= s^(α_i ^i+1)/qi (α_i+ 1)^qi1

Z si

0

(s_i−τ_i)^βiΦ_i(τ_i)^pidτ_i ¹

pi .

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

n

Y

i=1

s^w_iⁱ

!_Ωn¹

≤ 1

Ω_n

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 pi}¹

forr = (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=1ωis^(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, ¹

pi ds_i

#

≤ 1

Qn

i=1(α_i+ 1)^qi1

n

Y

i=1

x

1 qi

i

Z xi

0

Z si

0

(s_i−τ_i)^βiΦ_i(τ_i)^pidτ_i,

ds_i ¹

pi

= 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 2.2. In Theorem 2.1, settingΩ_n= 1, we have Theorem 1.1.

Corollary 2.3. Under the assumptions of Theorem 2.1, ifr >0, we have

Z x1

0

· · · Z xn

0

Qn i=1

u^(k_i ⁱ⁾s_i

h 1

Ωn

Pn

i=1ωis^(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 pi

Z xi

0

(x_i−s_i)^βi+1Φ_is^p_iⁱds_i r#_r·p¹

,

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 2.4. Letγ1 >0andγ2 <−1.Letωi >0, Pn

i=1ωi = Ωnand letsi >0, i= 1, . . . , n be 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 Ωn

n

X

i=1

ω_is_i

#γ1γ2Ωn

.

Using the fact thatx^−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

=



 1 Ω_n

n

X

i=1

ω_is^−γ_i ²

!−_γ¹

2





γ1γ2Ωn

=

"

1 Ω_n

n

X

i=1

ωis^−γ_i ²

#^−γ1Ωn

.

The proof of the lemma is complete.

Theorem 2.5. Under the assumptions of Theorem 2.1, ifγ₂ <−1,then 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

≤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 ¹

pi ,

whereV is given by (2.3).

Proof. Using the inequality of Lemma 2.4, for anyγ₁ >0andγ₂ <−1,we get

n

Y

i=1

s^ω_i^iγ1 ≤

"

1 Ω_n

n

X

i=1

ω_is^−γ_i ²

#−^γ1Ωn_γ

2

.

According to above inequality, we deduce that

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^−γ_i ²

#−W1

×

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

Theorem 2.1.

Corollary 2.6. Under the assumptions of Theorem 2.5, 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· · ·dsn

≤p^r·p1 V

n

Y

i=1

x

1 qi

i

" _n X

i=1

1 pi

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 anyAi ≥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 2.5.

In the following section we discuss some choice of the functionsΦ_i. 3. THEVARIOUSINEQUALITIES

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 ¹

pi ,

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 Theorem 2.1.

Corollary 3.2. Under the assumptions of Theorem 3.1, 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

(xi−si)^βi+1

u^(m_i ⁱ⁾(si)

pi

dsi

r#_r·p¹ , whereV₁is given by (3.2).

Theorem 3.3. Under the assumptions of Theorem 3.1, 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^−(αi+1)Ωn/γ2qiωi ds₁· · ·ds_n

≤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

, whereV₁is given by (3.2).

Proof. Inequality (3.3) is proved when we set

Φ_i(s_i) =

u^(m_i ⁱ⁾(s_i) (m_i−k_i−1)!

in Theorem 2.5.

Corollary 3.4. Under the assumptions of Theorem 3.3, 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 pi

Z xi

0

(x_i−s_i)^βi+1

u^(m_i ⁱ⁾(s_i)

pi

ds_i r#_r·p¹

.

We discuss a number of special cases of Theorem 3.1. Similar examples apply also to Corol- lary 3.2, Theorem 3.3 and Corollary 3.4.

Example 3.1. Ifa_i = 0andb_i = 1fori∈I,then Theorem 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^(q_i ^{imi−qiki−qi+1)/(qiωi)}iΩn ds1· · ·dsn

≤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

(mi −ki−1)!(qimi−qiki−qi+ 1)

1 qi

i.

Example 3.2. Ifa_i = 0, b_i = 1, q_i =n, p_i =n/(n−1), m_i =mandk_i =kfori∈I,then

Z x1

0

· · · Z xn

0

Qn i=1

u^(k_i ⁱ⁾(s_i) h 1

Ωn

Pn

i=1ω_is(nm−nk−n+1)/(nω_i) 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 .

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

Example 3.3. Ifa_i = 0andb_i = 1fori∈I,then Theorem 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_i ^{i−ki)/(qiωi)}iΩn ds₁· · ·ds_n

≤V3 n

Y

i=1

x

1 qi

i n

Y

i=1

Z xi

0

(xi−si)^mi−ki

u^(m_i ⁱ⁾(si)

pi

dsi

_pi¹ , where

V₃ = 1

Qn i=1

(mi−ki)!.

Example 3.4. Ifa_i = 1, b_i = 0, q_i = n, p_i =n/(n−1), m_i = mandk_i =kfori ∈ 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. Let p₁, p₂ ∈ R+. If we setn = 2, ω₁ = _p¹

1, ω₂ = _p¹

2, m_i = 1 andk_i = 0 for i= 1,2in Theorem 3.1, 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^(p₁ ¹⁻¹⁾+p₁s^(p₂²⁻¹⁾iΩ2 ds₁ds₂

≤x^(p₁ ^1−1)/p1x^(p₂^2−1)/p2 Z x1

0

(x₁−s₁)|u⁰₁(s₁)|^p1 ds_{1 p}¹

1

× Z x2

0

(x₂−s₂)|u⁰₂(s₂)|^p2 ds_{2 p}¹

2 .

If we setω1+ω2 = 1in Example 3.5, then we have [13, Theorem 2]. (The values ofai andbi

are irrelevant.)

REFERENCES

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

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

[3] G.H. HARDY, J.E. LITTLEWOODANDG. POLYA, Inequalities, Cambridge Univ. Press, London, 1952.

[4] YOUNG-HO KIM, Refinements and Extensions of an inequality, J. Math. Anal. Appl., 245 (2000), 628–632.

[5] V. LEVIN, On the two-parameter extension and analogue of Hilbert’s inequality, J. London Math.

Soc., 11 (1936), 119–124.

[6] G. MINGZE, On Hilbert’s inequality and its applications, J. Math. Anal. Appl., 212 (1997), 316–

323.

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

[8] D.S. MITRINOVI ´CANDJ.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.

[11] B.G. PACHPATTE, On some new inequalities similar to Hilbert’s inequality, 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.

[15] D.V. WIDDER, An inequality related to one of Hilbert’s, J. London Math. Soc., 4 (1929), 194–198.