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

_{and Wing-Sum Cheung}

China Jiliang University, Hangzhou 310018, China

2_{Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong}

Correspondence should be addressed to Chang-Jian Zhao,[email protected]

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 ms1asand

Bnnt1bt. Then

k

m1

r

n1

ApmBqn

mn ≤C

p, q, k, r

_k

m1

k−m1amApm−1

2 1/2 ×

_r

n1

r−n1bnBqn−1

2 1/2

,

where

Cp, 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

tn1bt1,...,tn.Then

k1

m11

· · · kn

mn1

r1

n11

· · ·rn

nn1

αβt1/β_Ap m1,...,mnB

q n1,...,nn m1· · ·mnβn1· · ·nnαt

≤Lk1, . . . , kn, r1, . . . , rn, p, q, α, β

×

k1

m11

· · · kn

mn1

n

j1

kj−mj1am1,...,mnA

p−1

m1,...,mn

β 1/β

× ⎛ ⎝r1

n11

· · ·rn

nn1

n

j1

rj−nj1bn1,...,nnB

q−1

n1,...,nn _α⎞

⎠ 1/α

,

2.3

where

Lk1, . . . , kn, r1, . . . , rn, p, q, α, βpqk1· · ·kn1/αr1· · ·rn1/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

Apm1,...,mn ≤p

m1

s11

· · ·mn

sn1

as1,...,snA

p−1

s1,...,sn. 2.6

Similarly, we have

Bnq1,...,nn ≤q

n1

t11

· · ·nn

tn1

bt1,...,tnB

q−1

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

a1/α_b1/β_≤ a

αt1/β

t1/α_b

β , 2.8

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

Apmi,...,mnB

q

ni,...,nn ≤pq _m

1

s11

· · ·mn

sn1

as1,...,snA

p−1

s1,...,sn

n1

t11

· · ·nn

tn1 bt1,...,tnB

q−1

t1,...,tn

≤pqm1, . . . , mn1/α

_m

1

s11

· · ·mn

sn1

as1,...,snA

p−1

s1,...,sn _β 1/β

×n1, . . . , nn1/β

_n

1

t11

· · ·nn

tn1

bt1,...,tnB

q−1

t1,...,tn α 1/α

≤pq

m1· · ·mn

αt1/β

n1· · ·nnt1/α

β

m1

s11

· · ·mn

sn1

as1,...,snA

p−1

s1,...,sn β 1/β

× _n

1

t11 . . .nn

tn1

bt1,...,tnB

q−1

t1,...,tn _α 1/α

.

2.9

Dividing both sides of2.9bym1· · ·mnβn1· · ·nnαt/αβt1/β,summing up overnifrom

1 tori i1,2, . . . , nfirst, then summing up overmifrom 1 toki 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

αβt1/β_Ap m1,...,mnB

q n1,...,nn m1· · ·mnβn1· · ·nnαt

≤pq

⎧ ⎨ ⎩

k1

m11

· · · kn

mn1

_m

1

s11

· · ·mn

sn1

as1,...,snA

p−1

s1,...,sn β 1/β

⎫ ⎬ ⎭

× ⎧ ⎨ ⎩

r1

n11

· · ·rn

nn1 _n

1

t11

· · ·nn

tn1

bt1,...,tnB

q−1

t1,...,tn α 1/α

⎫ ⎬ ⎭

≤pqk1· · ·kn1/α _k

1

m11

· · · kn

mn1 _m

1

s11

· · ·mn

sn1

as1,...,snA

p−1

s1,...,sn

_β 1/β

×r1· · ·rn1/β

_k

1

n11

· · ·rn

nn1 _n

1

t11

· · ·nn

tn1

bt1,...,tnB

q−1

t1,...,tn

Lk1, . . . , kn, r1, . . . , rn, p, q, α, β

× _k

1

s11

· · ·kn

sn1

as1,...,snA

p−1

s1,...,sn

βk1

m1s1

· · · kn

mnsn 1

1/β

× _r

1

t11

· · ·rn

tn1

bt1,...,tnB

q−1

t1,...,tn

αr1

n1t1

· · · rn

nntn 1

1/α

Lk1, . . . , kn, r1, . . . , rn, p, q, α, β

× ⎧ ⎨ ⎩

k1

s11

· · ·kn

sn1

n

j1

kj−sj1as1,...,snA

p−1

s1,...,sn β

⎫ ⎬ ⎭

1/β

× ⎧ ⎨ ⎩

r1

t11

· · ·rn

tn1

n

j1

rj−tj1bt1,...,tnB

q−1

t1,...,tn _α⎫⎬

⎭ 1/α

Lk1, . . . , kn, r1, . . . , rn, p, q, α, β

× ⎧ ⎨ ⎩

k1

m11

· · · kn

mn1

n

j1

kj−mj1am1,...,mnA

p−1

m1,...,mn _β⎫⎬

⎭ 1/β

× ⎧ ⎨ ⎩

r1

n11

· · ·rn

nn1

n

j1

rj−nj1bn1,...,nnB

q−1

n1,...,nn α

⎫ ⎬ ⎭

1/α

.

2.10

This completes the proof.

Remark 2.2. Takingαβnj2,2.3becomes

k1

m1

k2

m21

_r

1

n11

r2

n21

Apm1,m2B

q n1,n2 m1m2t−1/2n1n2t1/2

≤ 1

2pq

k1k2r1r2 _k

1

m1

k2

m21

k1−m11k2−m21

am1,m2A

p−1

m1,m2

2 1/2

× _r

1

n1

r2

n21

r1−n11r2−n21

bn1,n2B

p−1

n1,n2

2 1/2

.

2.11

Takingt1,and changing{am1,m2},{bn1,n2},{Am1,m2},and{Bn1,n2}into{am},{bn},{Am},and

{Bn},respectively, and with suitable changes,2.11reduces to Pachpatte1, inequality1.

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 nt1qt. Let φ and ψ be real-valued, nonnegative, convex, submultiplicative functions defined onR 0,∞.Then

k

m1

r

n1

φAmψBn

mn ≤Mk, r

_k

m1

k−m1

pmφ

_a

m

pm

2 1/2

× _r

n1

r−n1

qnφ

_b

n

qn

2 1/2

,

2.12

where

Mk, r 1 2

_k

m1 _φP

m

Pm

2 1/2r

n1 _φQ

n

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. DefinePm1,...,mn

_m1

s11· · ·

mn

sn1ps1,...,sn,andQn1,...,nn _n1

t11· · ·

nn

tn1qt1,...,tn.Letφ

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

Then

k1

m11

· · · kn

mn1

r1

n11

· · ·rn

nn1

αβt1/β_φA

m1,...,mnψBn1,...,nn m1· · ·mnβn1· · ·nnαt

≤Mk1, . . . , kn, r1, . . . , rn, α, β

× ⎧ ⎨ ⎩

k1

m11

· · · kn

mn1

n

j1

kj−mj1pm1,...,mnφ _a

m1,...,mn pm1,...,mn

_β⎫_⎬ ⎭

1/β

× ⎧ ⎨ ⎩

r1

n11

· · ·rn

nn1

n

j1

rj−nj1qn1,...,nnψa _b

n1,...,nn qn1,...,mn

α⎫⎬

⎭ 1/α

,

2.14

where

Mk1, . . . , kn, r1, . . . , rn, α, β

_k

1

m11

· · · kn

mn1 _φP

m1,...,mn Pm1,...,mn

α 1/αr1

n11

· · ·rn

nn1 _ψQ

n1,...,nn Qn1,...,nn

β 1/β

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

φAm1,...,mn φ

Pm1,...,mn _m1

s11. . .

_mn

sn1ps1,...,sn

as1,...,sn/ps1,...,sn _m1

s11· · ·

mn

sn1ps1,...,sn

≤φPm1,...,mnφ _m1

s11· · ·

mn

sn1ps1,···,sn

as1,...,sn/ps1,...,sn m1

s11· · ·

mn

sn1ps1,...,sn

≤ φPm1,...,mn Pm1,...,mn

m1

s11

· · ·mn

sn1 ps1,...,snφ

_a

s1,...,sn ps1,...,sn

≤ φPm1,...,mn Pm1,...,mn

m1· · ·mn1/α

_m

1

s11

· · ·mn

sn1

ps1,...,snφ _a

s1,...,sn ps1,...,sn

_β 1/β

.

2.16

Similarly,

φBn1,...,nn≤

φQn1,...,nn Qn1,...,nn

n1· · ·nn1/β _n

1

n11

· · ·nn

nn1

qt1,...,tnψ _b

t1,...,tn qt1,...,tn

α 1/α

. 2.17

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

a1/α_b1/β_≤ a

αt1/β

t1/α_b

β , 2.18

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

φAm1,...,mnφBn1,...,nn≤

m1· · ·mn

αt1/β

n1· · ·nnt1/α

β

× φPm1,...,mn Pm1,...,mn

_m

1

s11

· · ·mn

sn1

ps1,...,snφ _a

s1,...,sn ps1,...,sn

_β 1/β

× φQn1,...,nn Qn1,...,nn

_n

1

t11 . . .nn

tn1

qt1,...,tnψ _b

t1,...,tn qt1,...,tn

α 1/α

.

Dividing both sides of2.19bym1· · ·mnβn1· · ·nnαt/αβt1/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

αβt1/β_φAm

1,...,mnψBn1,...,nn m1. . . mnβn1. . . nnαt

≤ k1

m11 . . . kn

mn1 ⎛

⎝φPm1,...,mn Pm1,...,mn

_m

1

s11

· · ·mn

sn1

ps1,...,snφ _a

s1,...,sn ps1,...,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 _φP

m1,...,mn Pm1,...,mn

α 1/α

×

_k

1

m11

· · · kn

mn1 _m

1

s11

· · ·mn

sn1

ps1,...,snφ _a

s1,...,sn ps1,...,sn

_β 1/β

× _r

1

n11

· · ·rn

nn1 _ψQ

n1,...,nn Qn1,...,nn

β 1/β

× _r

1

n11

· · ·rn

nn1 _n

1

t11

· · ·nn

tn1

qt1,...,tnψ _b

t1,...,tn qt1,...,tn

α 1/α

Mk1, . . . , kn, r1, . . . , rn, α, β

× ⎧ ⎨ ⎩

k1

m11

· · · kn

mn1

n

j1

kj−mj1pm1,...,mnφ _a

m1,...,mn pm1,...,mn

β⎫⎬

⎭ 1/β

× ⎧ ⎨ ⎩

r1

n11

· · ·rn

nn1

n

j1

rj−nj1qn1,...,nnψ _b

n1,...,nn qn1,...,mn

α⎫⎬

⎭ 1/α

.

2.20

Remark 2.4. Takingαβnj2,2.14becomes

k1

m1

k2

m21

_r

1

n11

r2

n21

φAm1,m2ψBn1,n2 m1m2t−1/2n1n2t1/2

≤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 qn1,...,nn

2 1/2

,

2.21

where

Mk1, k2, r1, r2

_k

1

m11

k2

m21

_φP

m1,m2 Pm1,m2

2 1/2r1

n11

r2

n21

_ψQ

n1,n2 Qn1,n2

2 1/2

. 2.22

Takingt1,and changing{am1,m2},{bn1,n2},{Am1,m2},and{Bn1,n2}into{am},{bn},{Am},and

{Bn},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 Pm1,...,mn

m1

s11

· · ·mn

sn1

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

Bn1,...,n2

1 Qn1,...,nn

n1

t11

· · ·nn

tn1

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

2.23

formi 1,2, . . . , ki,and ni 1,2, . . . , ri i 1,2, . . . , n,where ki, ri i 1, . . . , n are natural numbers. Letφandψbe real-valued, nonnegative, convex functions defined onR 0,∞.Then

k1

m11

· · · kn

mn1

r1

n11

· · ·rn

nn1

αβt1/β_P

m1,...,mnQn1,...,nnφAm1,...,mnψBn1,...,nn m1· · ·mnβn1· · ·nnαt

k1· · ·kn1/αr1· · ·rn1/β

× ⎧ ⎨ ⎩

k1

m11

· · · kn

mn1

n

j1

kj−mj1pm1,...,mnφam1,...,mn _β⎫⎬

⎭ 1/β

× ⎧ ⎨ ⎩

r1

n11

· · ·rn

nn1

n

j1

rj−nj1qn1,...,nnψbn1,...,nn α

⎫ ⎬ ⎭

1/α

.

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

≤ _P 1

m1,...,mn

m1

s11

· · ·mn

sn1

ps1,...,snφas1,...,sn

≤ _P 1

m1,...,mn

m1· · ·mn1/α

_m

1

s11

· · ·mn

sn1

ps1,...,snφas1,...,sn β 1/β_,

2.25

ψBn1,...,nn ψ

1 Qn1,...,nn

n1

t11

· · ·nn

tn1

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

≤ _Q 1

n1,...,nn

n1

t11

· · ·nn

tn1

qt1,...,tnψbt1,...,tn

≤ _Q 1

n1,...,nn

n1· · ·nn1/β

_n

1

t11

· · ·nn

tn1

qt1,...,tnψbt1,...,tn α 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

PmQnφAmψBn

mn ≤

1 2kr

1/2

_k

m1

k−m1p_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

Theorem D. Let{am},{bn}be as defined in Theorem A. DefineAm 1/mms1as,andBn

1/nn_t1b_t,form 1, . . . , k,andn 1, . . . , r, wherek, r are natural numbers. Letφ andψ be real-valued, nonnegative, convex functions defined onR 0,∞.Then

k

m1

r

n1

mn

mnφAmψBn≤ 1 2kr

1/2

_k

m1

k−m1φam2

1/2

× _r

n1

r−n1ψb_n2

1/2

.

2.28

This is the new inequality of Pachpatte in [1, Theorem 3].

Acknowledgments

Research is supported by Zhejiang Provincial Natural Science Foundation of ChinaY605065, Foundation of the Education Department of Zhejiang Province of China20050392. Research is partially supported by the Research Grants Council of the Hong Kong SAR, ChinaProject no. HKU7016/07Pand a HKU Seed Grant forBasic Research.

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