Some New Hilbert’s Type Inequalities

Journal of Inequalities and Applications Volume 2009, Article ID 851360,10pages doi:10.1155/2009/851360

Research Article

Some New Hilbert’s Type Inequalities

Chang-Jian Zhao

¹

and Wing-Sum Cheung

²

1Department of Information and Mathematics Sciences, College of Science, China Jiliang University, Hangzhou 310018, China

2Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong

Correspondence should be addressed to Chang-Jian Zhao,chjzhao@163.com Received 25 December 2008; Accepted 24 April 2009

Recommended by Peter Pang

Some new inequalities similar to Hilbert’s type inequality involving series of nonnegative terms are established.

Copyrightq2009 C.-J. Zhao and W.-S. Cheung. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

In recent years, several authors 1–10 have given considerable attention to Hilbert’s type inequalities and their various generalizations. In particular, in1, Pachpatte proved some new inequalities similar to Hilbert’s inequality11, page 226involving series of nonnegative terms. The main purpose of this paper is to establish their general forms.

2. Main Results

In1, Pachpatte established the following inequality involving series of nonnegative terms.

Theorem A. Letp≥1, q≥1,and let{am}and{bn}be two nonnegative sequences of real numbers defined form 1, . . . , k,andn 1, . . . , r, wherek, r are natural numbers. LetAm _m

s1asand B_nn

t1b_t. Then

k m1

r n1

A^p_mB^q_n mn ≤C

p, q, k, r _k

m1

k−m1

a_mA^p−1_m 2 1/2

× _r

n1

r−n1

b_nB^q−1_n 2 1/2

, 2.1

where

C

p, q, k, r 1

2pqkr^1/2. 2.2

We first establish the following general form of inequality2.1.

Theorem 2.1. Letp ≥ 1, q ≥ 1, t > 0,and 1/α1/β 1, α > 1. Let {a_m1,...,mn},and{b_n1,...,nn} be positive sequences of real numbers defined for mi 1,2, . . . , ki, and ni 1,2, . . . , ri, where ki, ri i 1, . . . , n are natural numbers. Let Am1,...,mn _m1

s11· · ·_mn

sn1as1,...,sn, and Bn1,...,nn _n1

t11· · ·_nn

tn1b_t1,...,tn.Then

k1

m11

· · · ^kn

mn1 r1

n11

· · ·^rn

nn1

αβt^1/βA^p_m1,...,mnB_n^q_1,...,nn m1· · ·mnβn1· · ·nnαt

≤L

k1, . . . , k_n, r1, . . . , r_n, p, q, α, β

× k1

m11· · · ^kn

mn1

n j1

kj−mj1

am1,...,mnA^p−1_m1,...,mnβ ^1/β

×

⎛

⎝^r1

n11

· · ·^rn

nn1

n j1

rj−nj1

bn1,...,nnB_n^q−1_1,...,nnα⎞

⎠

1/α

,

2.3

where

L

k1, . . . , k_n, r1, . . . , r_n, p, q, α, β

pqk1· · ·k_n^1/αr1· · ·r_n^1/β. 2.4

Proof. By using the following inequalitysee12:

_n 1

m11

· · · ⁿⁿ

mn1

zm1,...,mn

p

≤pⁿ¹

m11

, . . . ,ⁿⁿ

mn1

zm1,...,mn

_m 1

k11

· · ·^mn

kn1

zk1,...,kn

p−1

, 2.5

wherep≥1 is a constant, andz_m1,...,mn ≥0, m_i1,2, . . . , k_i,i1,2, . . . , n, we obtain

A^p_m1,...,mn ≤p^m1

s11

· · ·^mn

sn1

a_s1,...,snA^p−1_s1,...,sn. 2.6

Similarly, we have

B_n^q_1,...,nn ≤qⁿ¹

t11

· · ·ⁿⁿ

tn1

b_t1,...,tnB^q−1_t1,...,tn. 2.7

From2.6and2.7, using H ¨older’s inequality13and the elementary inequality:

a^1/αb^1/β≤ a

αt^1/β t^1/αb

β , 2.8

where 1/α1/β1, α >1, b >0, a >0, andt >0,we have

A^p_mi,...,mnB^q_ni,...,nn ≤pq _m

1

s11

· · ·^mn

sn1

as1,...,snA^p−1_s1,...,sn ⁿ¹

t11

· · ·ⁿⁿ

tn1

bt1,...,tnB_t^q−1_1,...,tn

≤pqm1, . . . , m_n^1/α _m

1

s11

· · ·^mn

sn1

a_s1,...,snA^p−1_s1,...,snβ ^1/β

×n1, . . . , n_n^1/β _n

1

t11

· · ·ⁿⁿ

tn1

b_t1,...,tnB^q−1_t1,...,tnα ^1/α

≤pq

m1· · ·m_n

αt^1/β n1· · ·n_nt^1/α β

m1

s11

· · ·^mn

sn1

as1,...,snA^p−1s1,...,sn

_β ^1/β

× _n

1

t11

. . .ⁿⁿ

tn1

bt1,...,tnB^q−1_t1,...,tnα ^1/α .

2.9

Dividing both sides of2.9bym1· · ·mnβn1· · ·nnαt/αβt^1/β,summing up overnifrom 1 tor_i i1,2, . . . , nfirst, then summing up overm_ifrom 1 tok_i i1,2, . . . , n, using again H ¨older’s inequality, then interchanging the order of summation, we obtain

k1

m11

· · · ^kn

mn1 r1

n11

· · ·^rn

nn1

αβt^1/βA^p_m1,...,mnB_n^q_1,...,nn m1· · ·mnβn1· · ·nnαt

≤pq

⎧⎨

⎩

k1

m11

· · · ^kn

mn1

_m 1

s11

· · ·^mn

sn1

as1,...,snA^p−1s1,...,sn

_β ^1/β⎫

⎬

⎭

×

⎧⎨

⎩

r1

n11

· · ·^rn

nn1

_n 1

t11

· · ·ⁿⁿ

tn1

bt1,...,tnB^q−1_t1,...,tnα ^1/α⎫

⎬

⎭

≤pqk1· · ·kn^1/α _k

1

m11

· · · ^kn

mn1

_m 1

s11

· · ·^mn

sn1

as1,...,snA^p−1_s1,...,snβ ^1/β

×r1· · ·r_n^1/β _k

1

n11

· · ·^rn

nn1

_n 1

t11

· · ·ⁿⁿ

tn1

b_t1,...,tnB_t^q−1_1,...,tnα ^1/α

L

k1, . . . , kn, r1, . . . , rn, p, q, α, β

× _k

1

s11

· · ·^kn

sn1

a_s1,...,snA^p−1_{s1,...,snβ k} 1

m1s1

· · · ^kn

mnsn

1

1/β

× _r

1

t11

· · ·^rn

tn1

bt1,...,tnB^q−1_{t1,...,tnα r} 1

n1t1

· · · ^rn

nntn

1

1/α

L

k1, . . . , k_n, r1, . . . , r_n, p, q, α, β

×

⎧⎨

⎩

k1

s11

· · ·^kn

sn1

n j1

kj−sj1

as1,...,snA^p−1s1,...,sn

_β⎫

⎬

⎭

1/β

×

⎧⎨

⎩

r1

t11

· · ·^rn

tn1

n j1

r_j−t_j1

b_t1,...,tnB_t^q−1_1,...,tnα⎫

⎬

⎭

1/α

L

k1, . . . , kn, r1, . . . , rn, p, q, α, β

×

⎧⎨

⎩

k1

m11

· · · ^kn

mn1

n j1

kj−mj1

am1,...,mnA^p−1_m1,...,mnβ⎫

⎬

⎭

1/β

×

⎧⎨

⎩

r1

n11

· · ·^rn

nn1

n j1

rj−nj1

bn1,...,nnB_n^q−1_1,...,nnα⎫

⎬

⎭

1/α

.

2.10

This completes the proof.

Remark 2.2. Takingαβnj2,2.3becomes

k1

m1 k2

m21

_r 1

n11 r2

n21

A^p_m1,m2B^q_n1,n2 m1m2t^−1/2n1n2t^1/2

≤ 1 2pq

k1k2r1r2

_k 1

m1 k2

m21

k1−m11k2−m21

am1,m2A^p−1_m1,m22 ^1/2

× _r

1

n1 r2

n21

r1−n11r2−n21

b_n1,n2B_n^p−1_1,n22 ^1/2 .

2.11

Takingt1,and changing{a_m1,m2},{b_n1,n2},{A_m1,m2},and{B_n1,n2}into{a_m},{b_n},{A_m},and {B_n},respectively, and with suitable changes,2.11reduces to Pachpatte1, inequality1.

In1, Pachpatte also established the following inequality involving series of nonneg- ative terms.

Theorem B. Let{am},{bn}, Am, Bn be as defined in Theorem A. Let {pm}and {qn} be positive sequences form1, . . . , k,andn1, . . . , r,wherek, r are natural numbers. DefineP_{m m}

s1p_s, and Qn _n

t1qt. Let φ and ψ be real-valued, nonnegative, convex, submultiplicative functions defined onR 0,∞.Then

k m1

r n1

φA_mψB_n

mn ≤Mk, r

_k

m1

k−m1

p_mφ a_m

p_m

₂ ^1/2

× _r

n1

r−n1

qnφ bn

q_n

2 1/2

,

2.12

where

Mk, r 1 2

_k

m1

φPm Pm

2 1/2 _r

n1

φQn Qn

2 1/2

. 2.13

Inequality2.12can also be generalized to the following general form.

Theorem 2.3. Let{a_m1,...,mn},{b_n1,...,nn},α, β, t, A_m1,...,mn,andB_n1,...,nn be as defined inTheorem 2.1.

Let{pm1,...,mn}and {qn1,...,nn}be positive sequences formi 1,2, . . . , ki,and ni 1,2, . . . , ri i 1,2, . . . , n. DefineP_{m1,...,mn m1}

s11· · ·_mn

sn1p_s1,...,sn,andQ_{n1,...,nn n1}

t11· · ·_nn

tn1q_t1,...,tn.Letφ and ψ be real-valued, nonnegative, convex, submultiplicative functions defined onR 0,∞.

Then

k1

m11

· · · ^kn

mn1 r1

n11

· · ·^rn

nn1

αβt^1/βφA_m1,...,mnψB_n1,...,nn m1· · ·m_nβn1· · ·n_nαt

≤M

k1, . . . , kn, r1, . . . , rn, α, β

×

⎧⎨

⎩

k1

m11

· · · ^kn

mn1

n j1

k_j−m_j1

p_m1,...,mnφ

a_m1,...,mn p_m1,...,mn

_β⎫

⎬

⎭

1/β

×

⎧⎨

⎩

r1

n11

· · ·^rn

nn1

n j1

rj−nj1

qn1,...,nnψa

b_n1,...,nn qn1,...,mn

α⎫

⎬

⎭

1/α

,

2.14

where

M

k1, . . . , k_n, r1, . . . , r_n, α, β

_k 1

m11

· · · ^kn

mn1

φP_m1,...,mn P_m1,...,mn

_α ^1/α _r 1

n11

· · ·^rn

nn1

ψQ_n1,...,nn Q_n1,...,nn

_β ^1/β

. 2.15

Proof. By the hypotheses, Jensen’s inequality, and H ¨older’s inequality, we obtain

φA_m1,...,mn φ

Pm1,...,mn

_m1

s11. . ._mn

sn1ps1,...,sn

as1,...,sn/ps1,...,sn

_m1

s11· · ·_mn

sn1ps1,...,sn

≤φPm1,...,mnφ _m1

s11· · ·_mn

sn1p_s1,···,sn

a_s1,...,sn/p_{s1,...,sn m1}

s11· · ·_mn

sn1p_s1,...,sn

≤ φPm1,...,mn Pm1,...,mn

m1

s11

· · ·^mn

sn1

p_s1,...,snφ

as1,...,sn

ps1,...,sn

≤ φP_m1,...,mn

P_m1,...,mn m1· · ·m_n^1/α _m

1

s11

· · ·^mn

sn1

p_s1,...,snφ

as1,...,sn

p_s1,...,sn

_β ^1/β .

2.16

Similarly,

φBn1,...,nn≤ φQn1,...,nn

Q_n1,...,nn n1· · ·nn^1/β _n

1

n11

· · ·ⁿⁿ

nn1

qt1,...,tnψ

b_t1,...,tn q_t1,...,tn

_α ^1/α

. 2.17

By2.16and2.17, and using the elementary inequality:

a^1/αb^1/β≤ a

αt^1/β t^1/αb

β , 2.18

where 1/α1/β1, α >1, b >0, a >0,andt >0,we have

φA_m1,...,mnφB_n1,...,nn≤

m1· · ·m_n

αt^1/β n1· · ·n_nt^1/α β

× φPm1,...,mn Pm1,...,mn

_m 1

s11

· · ·^mn

sn1

p_s1,...,snφ

as1,...,sn

ps1,...,sn

_β 1/β

× φQ_n1,...,nn Q_n1,...,nn

_n 1

t11

. . .ⁿⁿ

tn1

q_t1,...,tnψ

bt1,...,tn

q_t1,...,tn

_α ^1/α .

2.19

Dividing both sides of2.19bym1· · ·mnβn1· · ·nnαt/αβt^1/β,and summing up overni

from 1 tor_i i1,2, . . . , nfirst, then summing up overm_ifrom 1 tok_i i1,2, . . . , n, using again inverse H ¨older’s inequality, and then interchanging the order of summation, we obtain

k1

m11

· · · ^kn

mn1 r1

n11

· · ·^rn

nn1

αβt^1/βφAm1,...,mnψBn1,...,nn m1. . . mnβn1. . . nnαt

≤ ^k1

m11

. . . ^kn

mn1

⎛

⎝φPm1,...,mn P_m1,...,mn

_m 1

s11

· · ·^mn

sn1

ps1,...,snφ

a_s1,...,sn p_s1,...,sn

_β ^1/β⎞

⎠

×^r1

n11

· · ·^rn

nn1

⎛

⎝φQn1,...,nn Qn1,...,nn

_n 1

n11

· · ·ⁿⁿ

nn1

qt1,...,tnψ

b_t1,...,tn qt1,...,tn

α 1/α⎞

⎠

≤ _k

1

m11

· · · ^kn

mn1

φPm1,...,mn Pm1,...,mn

α 1/α

× _k

1

m11

· · · ^kn

mn1

_m 1

s11

· · ·^mn

sn1

p_s1,...,snφ

a_s1,...,sn p_s1,...,sn

_β 1/β

× _r

1

n11

· · ·^rn

nn1

ψQn1,...,nn Qn1,...,nn

β 1/β

× _r

1

n11

· · ·^rn

nn1

_n 1

t11

· · ·ⁿⁿ

tn1

qt1,...,tnψ

b_t1,...,tn q_t1,...,tn

_{α 1/α}

M

k1, . . . , k_n, r1, . . . , r_n, α, β

×

⎧⎨

⎩

k1

m11

· · · ^kn

mn1

n j1

kj−mj1

pm1,...,mnφ

am1,...,mn

pm1,...,mn

_β⎫

⎬

⎭

1/β

×

⎧⎨

⎩

r1

n11

· · ·^rn

nn1

n j1

r_j−n_j1

q_n1,...,nnψ

bn1,...,nn

q_{n1,...,mn α}⎫

⎬

⎭

1/α

.

2.20 The proof is complete.

Remark 2.4. Takingαβnj2,2.14becomes

k1

m1 k2

m21

_r 1

n11 r2

n21

φAm1,m2ψBn1,n2 m1m2t^−1/2n1n2t^1/2

≤Mk1, k2, r1, r2

× _k

1

m1 k2

m21

k1−m11k2−m21

pm1,m2φ

a_m1,...,mn pm1,...,mn

2 1/2

× _r

1

n1 r2

n21

r1−n11r2−n21

qn1,n2ψ

b_n1,...,nn q_n1,...,nn

2 1/2

,

2.21

where

Mk1, k2, r1, r2

_k 1

m11 k2

m21

φP_m1,m2 P_m1,m2

2 1/2 _r 1

n11 r2

n21

ψQ_n1,n2 Q_n1,n2

2 1/2

. 2.22

Takingt1,and changing{a_m1,m2},{b_n1,n2},{A_m1,m2},and{B_n1,n2}into{a_m},{b_n},{A_m},and {B_n},respectively, and with suitable changes,2.21reduces to Pachpatte1, Inequality7.

Theorem 2.5. Let{am1,...,mn},{bn1,...,nn},{pm1,...,mn},{qn1,...,nn},Pm1,...,mn,andQn1,...,nn,α, β, t,be as defined inTheorem 2.3. Define

Am1,...,mn 1 P_m1,...,mn

m1

s11

· · ·^mn

sn1

ps1,...,snas1,...,sn, Bn1,...,n2 1

Q_n1,...,nn

n1

t11

· · ·ⁿⁿ

tn1

qt1,...,tnbt1,...,tn,

2.23

form_i 1,2, . . . , k_i,and n_i 1,2, . . . , r_i i 1,2, . . . , n,where k_i, r_i i 1, . . . , n are natural numbers. Letφandψbe real-valued, nonnegative, convex functions defined onR 0,∞.Then

k1

m11

· · · ^kn

mn1 r1

n11

· · ·^rn

nn1

αβt^1/βP_m1,...,mnQ_n1,...,nnφA_m1,...,mnψB_n1,...,nn m1· · ·mnβn1· · ·nnαt

k1· · ·k_n^1/αr1· · ·r_n^1/β

×

⎧⎨

⎩

k1

m11

· · · ^kn

mn1

n j1

kj−mj1

pm1,...,mnφam1,...,mn_β⎫

⎬

⎭

1/β

×

⎧⎨

⎩

r1

n11

· · ·^rn

nn1

n j1

r_j−n_j1

q_n1,...,nnψb_n1,...,nnα⎫

⎬

⎭

1/α

.

2.24

Proof. By the hypotheses, Jensen’s inequality, and H ¨older’s inequality, it is easy to observe that

φAm1,...,mn φ 1

Pm1,...,mn

m1

s11

· · ·^mn

sn1

ps1,...,snas1,...,sn

≤ 1

P_m1,...,mn

m1

s11

· · ·^mn

sn1

ps1,...,snφas1,...,sn

≤ 1

P_m1,...,mnm1· · ·m_n^1/α _m

1

s11

· · ·^mn

sn1

p_s1,...,snφa_s1,...,snβ ^1/β ,

2.25

ψBn1,...,nn ψ 1

Qn1,...,nn

n1

t11

· · ·ⁿⁿ

tn1

qt1,...,tnbt1,...,tn

≤ 1

Q_n1,...,nn

n1

t11

· · ·ⁿⁿ

tn1

qt1,...,tnψbt1,...,tn

≤ 1

Q_n1,...,nnn1· · ·n_n^1/β _n

1

t11

· · ·ⁿⁿ

tn1

q_t1,...,tnψb_t1,...,tnα ^1/α .

2.26

Proceeding now much as in the proof of Theorems2.1and2.3, and with suitable modifica- tions, it is not hard to arrive at the desired inequality. The details are omitted here.

Remark 2.6. In the special case wherej1, t1, αβ2,andn1,Theorem 2.5reduces to the following result.

Theorem C. Let {a_m},{b_n},{p_m},{q_n}, P_m, Q_n be as defined in Theorem B. Define A_m 1/P_mm

s1p_sa_s,andB_n 1/Q_nn

t1q_tb_t form 1, . . . , k,andn 1, . . . , r, wherek, r are natural numbers. Letφandψbe real-valued, nonnegative, convex functions defined onR 0,∞.

Then

k m1

r n1

P_mQ_nφA_mψB_n

mn ≤ 1

2kr^1/2 _k

m1

k−m1

p_mφa_m2 1/2

× _r

n1

r−n1qnψbn²

1/2

.

2.27

This is the new inequality of Pachpatte in [1, Theorem 4].

Remark 2.7. Takingj 1, t1, pm1, qn1, αβ2,andn1 inTheorem 2.5, and in view ofP_mm, Q_nn, we obtain the following theorem.