Some Refinements of Hilbert-Pachpatte type integral Inequalities

Some Refinements of Hilbert-Pachpatte type integral Inequalities

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Dragomir, Sever S and Kim, Young-Ho (2001) Some Refinements of Hilbert- Pachpatte type integral Inequalities. RGMIA research report collection, 4 (4).

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TYPE INTEGRAL INEQUALITIES

Sever S. Dragomir and Young-Ho Kim

Abstract. In this paper, we obtain an extension of a multivariable integral inequal- ity of Hilbert-Pachpatte type. By specializing the upper estimate functions in the hypothesis and the parameters, we obtain many special cases.

1. INTRODUCTION

Hilbert’s double series theorem [3, p.226] was proved first by Hilbert in his lec- tures on integral equations. The determination of the constant, the integral ana- logue, 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]). In particular, in [10-14] the present 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 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 u_i ∈ C^mi([0, xi]) for i∈I. If

|u^(k_i ⁱ⁾(si)|≤

Z (si) 0

(si−τi)^mi−ki−1Φi(τi)dτi, si ∈[0, xi], i ∈I,

1991Mathematics Subject Classification. 26 D 15.

Key words and phrases. Hilbert’s inequality, Hilbert-Pachpatte type Inequality, H¨older’s in- equality, Jensen inequality.

Typeset byAMS-TEX 1

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 Yn

i=1

x^1/q_i ⁱ Yn

i=1

’Z _xi

0

(xi−si)^βi+1Φi(si)^pidsi

“1/pi

,

where

U = 1

Qn

i=1[(α_i+ 1)^1/qi(β_i+ 1)^1/pi].

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. MAIN RESULTS

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 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₊,1/p_i+ 1/q_i = 1

p, q 1/p=

Xn

i=1

(1/p_i),1/q_i = Xn

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, Xn

i=1

ω_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

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 H¨older exponents to be used in ap- plications of H¨older’s inequality, and the coefficientsai, bi will be used in exponents to factorize integrands. The coefficientsω_i will act as weights in applications of the geometric-arithmetic mean inequality. The coefficients αi and βi arise naturally in the derivation of the inequalities.

Our main results are given in the following theorems.

Theorem 2.1. Let u_i∈C^mi([0, x_i]) for i∈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

Z x1

0 · · · Z xn

0

ΩnQn

i=1 |u^(k_i ⁱ⁾(si)| Pn

i=1ωis^(α_i ^{i+1)Ωn/(qiωi)} ds₁· · ·ds_n

≤V Yn

i=1

x^1/q_i ⁱ Yn

i=1

’Z xi

0

(x_i−s_i)^βi+1Φ_i(s_i)^pids_i

“1/pi

, (2.2)

where

(2.3) V =

’Yn i=1

[(α_i+ 1)^1/qi(β_i+ 1)^1/pi]

“−1

.

Proof. Factorize the integration 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 H¨older’s inequality [9, p.106]. Then

|u^(k_i ⁱ⁾(si)| ≤

’Z (si) 0

(si−τi)^{(ai+biqi)(mi−ki−1)}dτi,

“1/qi

×

’Z (si) 0

(si−τi)^{ai(mi−ki−1)}Φi(τi)^pidτi,

“1/pi

= s^(α_i ^i+1)/qi (αi+ 1)^1/qi

’Z (si) 0

(s_i−τ_i)^βiΦ_i(τ_i)^pidτ_i,

“1/pi

.

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

’Yn i=1

s^w_iⁱ

“1/Ωn

≤

’ 1 Ωn

Xn

i=1

w_is^r_i

“1/r

for r >0, we have Yn

i=1

|u^(k_i ⁱ⁾(si)|

≤W 1 Ω_n

Xn

i=1

ωis^(α_i ^{i+1)Ωn/(qiωi)} Yn

i=1

’Z _(si)

0

(si−τi)^βiΦi(τi)^pidτi,

“1/pi

,

where

W =

’Yn i=1

(α_i+ 1)^qi

“−1

.

In the following estimate we apply H¨older’s inequality and, at the end, change the order of integration:

Z x1

0 · · · Z xn

0

Ω_nQn

i=1 |u^(k_i ⁱ⁾(s_i)| Pn

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

≤W Yn

i=1

”Z _(xi)

0

’Z _(si)

0

(s_i−τ_i)^βiΦi(τ_i)^pidτi,

“1/pi

ds_i

•

≤W Yn

i=1

x^1/q_i ^I

”Z _(xi)

0

’Z _(si)

0

(si−τi)^βiΦi(τi)^pidτi,

“ dsi

•1/pi

= W

Qn

i=1(β_i+ 1)^1/pi Yn

i=1

x^1/q_i ⁱ Yn

i=1

’Z _xi

0

(xi−si)^βi+1Φi(si)^pidsi

“1/pi

.

This proves the theorem.

Remark. In Theorem 2.1, setting Ωn = 1, we have Theorem 1.1.

Corollary 2.2. Under the assumptions of Theorem 2.1, if r >0, we have Z x1

0 · · · Z xn

0

Ω_nQn

i=1 |u^(k_i ⁱ⁾(s_i)| Pn

i=1ωis^(α_i ^{i+1)Ωn/(qiωi)} ds₁· · ·ds_n

≤p^1/r·pV Yn

i=1

x^1/q_i ⁱ

’Xn i=1

1 p_i

’Z xi

0

(x_i−s_i)^βi+1Φ_i(s_i)^pids_i

“r“1/r·p

,

Proof. By the inequality of means, for any A_i ≥0 and r >0, we obtain Yn

i=1

A^1/p_i ⁱ ≤

’ p

Xn

i=1

1 p_iA^r_i

“1/r·p

.

The corollary then follows from the preceding theorem.

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

i=1ω_i = Ω_n and let s_i >

0, i= 1, . . . , n be real numbers. Then Yn

i=1

s^ω_i^iγ1γ2 ≥

’ 1 Ω_n

Xn

i=1

ω_is^−γ_i ²

“−γ1Ωn

.

Proof. By the inequality of means, for any γ₁ >0 and γ₂ <−1, Yn

i=1

s^ω_i^iγ1γ2 ≥

’ 1 Ω_n

Xn

i=1

ωisi

“γ1γ2Ωn

.

Using the fact that x^−1/γ2 is concave and using the Jensen inequality and we have that

’ 1 Ω_n

Xn

i=1

ω_is_i

“γ1γ2Ωn

=

’ 1 Ω_n

Xn

i=1

ω_if(s^−γ_i ²)

“γ1γ2Ωn

≥

’ f

’ 1 Ω_n

Xn

i=1

ω_is^−γ_i ²

““γ1γ2Ωn

=

’’ 1 Ωn

Xn

i=1

ω_is^−γ_i ²

“−1/γ2“γ1γ2Ωn

=

’ 1 Ω_n

Xn

i=1

ω_is^−γ_i ²

“−γ1Ωn

The proof of Lemma is complete.

Theorem 2.4. Under the assumptions of Theorem 2.1, if γ₂ <−1, then Z x1

0 · · · Z xn

0

Qn

i=1 |u^(k_i ⁱ⁾(s_i)|

 1 Ωn

Pn

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

≤V Yn

i=1

x^1/q_i ⁱ Yn

i=1

’Z _xi

0

(xi−si)^βi+1Φi(si)^pidsi

“1/pi

,

where V is given by (2.3).

Proof. By the inequality of Lemma 2.3, for any γ₁ >0 and γ₂ <−1, Yn

i=1

s^ω_i^iγ1 ≤

’ 1 Ω_n

Xn

i=1

ω_is^−γ_i ²

“−γ1Ωn/γ2

we get Yn

i=1

|u^(k_i ⁱ⁾(si)|

≤W

’ 1 Ω_n

Xn

i=1

ω_is^−γ_i ²

“−W1Yn i=1

’Z _(si)

0

(s_i−τ_i)^βiΦ_i(τ_i)^pidτ_i,

“1/pi

,

where

W =

’Yn i=1

(αi+ 1)^qi

“−1

, W1 = (αi+ 1)Ωn

γ₂q_iω_i . The Theorem then follows from the preceding Theorem 2.1.

Corollary 2.5. Under the assumptions of Theorem 2.4, if r >0, we have Z x1

0 · · · Z xn

0

Qn

i=1 |u^(k_i ⁱ⁾(s_i)|

 1 Ωn

Pn

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

≤p^1/r·pV Yn

i=1

x^1/q_i ⁱ

’Xn i=1

1 p_i

’Z xi

0

(x_i−s_i)^βi+1Φ_i(s_i)^pids_i

“r“1/r·p

,

where V is given by (2.3).

Proof. By the inequality of means, for any Ai ≥0 and r >0, we obtain Yn

i=1

A^1/p_i ⁱ ≤

’ p

Xn

i=1

1 p_iA^r_i

“1/r·p

.

The corollary then follows from the preceding Theorem 2.4.

In the following section we discuss some choice of the functions Φ_i. 3. THE VARIOUS INEQUALITIES

Theorem 3.1. Let u_i ∈C^mi([0, x_i]) be such that u^(j)_i (0) = 0 for j ∈ {0, . . . , m_i− 1}, i∈I. Then

Z x1

0 · · · Z xn

0

ΩnQn

i=1 |u^(k_i ⁱ⁾(si)| Pn

i=1ωis^(α_i ^{i+1)Ωn/(qiωi)} ds₁· · ·ds_n

≤V₁ Yn

i=1

x^1/q_i ⁱ Yn

i=1

’Z xi

0

(x_i−s_i)^βi+1 |u^(m_i ⁱ⁾(s_i)|^pi ds_i

“1/pi

, (3.1)

where

(3.2) V₁ =

’Yn i=1

(m_i−k_i−1)!(α_i+ 1)^1/qi(β_i+ 1)^1/pi‘“⁻¹ .

Proof. By [12, Eq. (7)],

u^(k_i ⁱ⁾(s_i) = 1 (m_i−k_i−1)!

Z si

0

(s_i−τ_i)^mi−ki−1u^m_i ⁱ(τ_i)dτ_i.

Inequality (3.1) is proved when we set

Φi(si) = |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, if r >0, we have Z x1

0 · · · Z xn

0

ΩnQn

i=1 |u^(k_i ⁱ⁾(si)| Pn

i=1ωis^(α_i ^{i+1)Ωn/(qiωi)} ds₁· · ·ds_n

≤p^1/r·pV₁ Yn

i=1

x^1/q_i ⁱ

’Xn i=1

1 pi

’Z xi

0

(x_i−s_i)^βi+1 |u^(m_i ⁱ⁾(s_i)|^pi ds_i

“r“1/r·p

,

where V1 is given by (3.2).

Theorem 3.3. Under the assumptions of Theorem 3.1, if γ2 <−1, then Z x1

0 · · · Z xn

0

Qn

i=1 |u^(k_i ⁱ⁾(s_i)|

 1 Ωn

Pn

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

≤V1

Yn

i=1

x^1/q_i ⁱ Yn

i=1

’Z _xi

0

(xi−si)^βi+1 |u^(m_i ⁱ⁾(si)|^pi dsi

“1/pi

, (3.3)

where V₁ is 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 2.4.

Corollary 3.4. Under the assumptions of Theorem 3.3, if r >0, we have Z x1

0 · · · Z xn

0

Qn

i=1 |u^(k_i ⁱ⁾(s_i)|

 1 Ωn

Pn

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

≤p^1/r·pV₁ Yn

i=1

x^1/q_i ⁱ

’Xn i=1

1 pi

’Z xi

0

(x_i−s_i)^βi+1 |u^(m_i ⁱ⁾(s_i)|^pi ds_i

“r“1/r·p

.

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

Example 3.1. If a_i = 0 and b_i = 1 for i∈I, then Theorem 3.1 becomes Z x1

0 · · · Z xn

0

Ω_nQn

i=1 |u^(k_i ⁱ⁾(s_i)| Pn

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

≤V₂ Yn

i=1

x^1/q_i ⁱ Yn

i=1

’Z xi

0

(x_i−s_i)|u^(m_i ⁱ⁾(s_i)|^pi ds_i

“1/pi

,

where

V₂ =

’Yn i=1

(m_i−k_i−1)!(q_im_i−q_ik_i−q_i+ 1)^1/qi‘“⁻¹ .

Example 3.2. If a_i = 0, b_i = 1, q_i = n, p_i = n/(n−1), m_i = m and k_i = k for i∈I, then

Z x1

0 · · · Z xn

0

Ω_nQn

i=1 |u^(k_i ⁱ⁾(s_i)| Pn

i=1ωis(nm−nk−n+1)Ωn/(mωi) i

ds₁· · ·ds_n

≤

√nx₁· · ·x_n

€(m−k−1)!n

(nm−nk−n+ 1)

× Yn

i=1

’Z _xi

0

(x_i−s_i)|u^(m)_i (s_i)|^n/(n−1) dsi

“(n−1)/n

.

For q=p =n= 2 and ω_i = 1/n this is [12, Theorem 1]. Setting q =p= 2, k = 0, n= 1 and ω_i = 1/n, we recover the result of [14].

Example 3.3. If a_i = 0 and b_i = 1 for i∈I, then Theorem 3.1 becomes Z x1

0 · · · Z xn

0

Ω_nQn

i=1 |u^(k_i ⁱ⁾(s_i)| Pn

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

≤V₃ Yn

i=1

x^1/q_i ⁱ Yn

i=1

’Z xi

0

(x_i−s_i)^mi−ki |u^(m_i ⁱ⁾(s_i)|^pi ds_i

“1/pi

,

where

V₃ =

’Yn i=1

€(m_i−k_i)!“−1

.

Example 3.4. If a_i = 1, b_i = 0, q_i = n, pi = n/(n−1), m_i = m and k_i = k for i∈I. Then (3.1) becomes

Z x1

0 · · · Z xn

0

Ω_nQn

i=1 |u^(k_i ⁱ⁾(s_i)| Pn

i=1ω_is^(m−k)Ω_i ^n/(nωi)ds1· · ·dsn

≤

√nx₁· · ·x_n

€(m−k)!n

Yn

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 set n = 2, ω₁ = 1/p₁, ω₂ = 1/p₂, m_i = 1 and ki = 0 for i = 1,2 in Theorem 3.1, then by our assumptions q1 = (p1 −1)/p1, q₂ = (p₂−1)/p₂, and we obtain

Z x1

0

Z x2

0

Ω₂ |u₁(s₁)||u₂(s₂)|

p₂s^Ω₁^2(p1−1)+p₁s^Ω₂^2(p2−1) ds₁ds₂

≤ x^(p₁ ^1−1)/p1x^(p₂ ^2−1)/p2 p₁p₂

’Z x1

0

(x₁−s₁)|u⁰₁(s₁)|^p1 ds₁

“1/p1

×

’Z x2

0

(x₂−s₂)|u⁰₂(s₂)|^p2 ds₂

“1/p2

.

If we set ω₁+ω₂ = 1 in Example 3.5, then we have [13, Theorem 2]. (The values of a_i and b_i are irrelevant.)

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School of Communications and Informatics Victoria University of Technology PO Box 14428, MCMC Melbourne, Victoria 8001, Australia

E-mail address: sever.dragomir@vu.edu.au

Department of Applied Mathematics Brain Korea 21 Project Corps Changwon National University, Changwon 641-773, Korea

E-mail address: yhkim@sarim.changwon.ac.kr