AN IMPROVEMENT OF SOME INEQUALITIES SIMILAR TO HILBERT’S INEQUALITY

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AN IMPROVEMENT OF SOME INEQUALITIES SIMILAR TO HILBERT’S INEQUALITY

YOUNG-HO KIM (Received 7 March 2001)

Abstract.We give an improvement of some inequalities similar to Hilbert’s inequality involving series of nonnegative terms. The integral analogies of the main results are also given.

2000 Mathematics Subject Classification. 26D15.

1. Introduction. It is well known that the following Hilbert’s double series inequal- ity (see [3, page 226]) plays an important role in many branches of mathematics.

Theorem1.1. Ifp >1,p=p/(p−1)and

a^pm≤A,

b^pn≤B, the summations running1to∞, then

ambn

m+n< π

sin(π /p)A^1/pB^1/p, (1.1) unless all the sequence{am}or{bn}is null.

The integral analogue of Hilbert’s inequality can be stated as follows (see [3, page 226]).

Theorem1.2. Ifp >1,p=p/(p−1)and _∞

0

f^p(x)dx≤F , _∞

0

g^p(y)dy≤G, (1.2)

then _∞

0

_∞

0

f (x)g(y)

x+y dx dy < π

sin(π /p)F^1/pG^1/p, (1.3) unlessf≡0org≡0.

These two theorems were studied extensively and numerous variants, generaliza- tions, and extensions appeared in the literature, see [1,2,3,5,6,9] and the references therein.

Recently, Pachpatte [9] gave new inequalities similar to Hilbert’s inequalities given in the above theorems, involving a series of nonnegative terms as follows.

Theorem1.3. Letp≥1, q≥1, and let{am}and{bn}be two nonnegative sequences of real numbers defined form=1,2, . . . , kandn=1,2, . . . , r ,wherek,rare the natural

numbers and defineAm=m

s=1as,Bn=n

t=1bt. Then k

m=1

r n=1

A^pmBn^q

m+n≤C(p, q, k, r )



^k

m=1

(k−m+1) A^p−1m am

2





1/2

×



^r

n=1

(r−n+1) Bn^q−1bn

2





1/2

,

(1.4)

unless{am}or{bn}is null, where

C(p, q, k, r )=1 2pq

kr . (1.5)

An integral analogue ofTheorem 1.3is given in the following theorem.

Theorem1.4. Letp≥1,q≥1andf (σ )≥0, g(τ)≥0forσ ∈(0, x),τ∈(0, y), wherex,yare positive real numbers and defineF (s)=s

0f (σ )dσandG(t)=t

0g(τ)dτ, fors∈(0, x),t∈(0, y). Then

x 0

y 0

F^p(s)G^q(t)

s+t ds dt≤D(p, q, x, y) x

0(x−s)

F^p−1(s)f (s)2

ds 1/2

× y

0

(y−t)

G^q−1(t)g(t)2

dt 1/2

,

(1.6)

unlessf≡0org≡0, where

D(p, q, x, y)=1 2pq

xy. (1.7)

In this paper, we give an improvement of the inequalities given in Theorems1.3and 1.4similar to Hilbert’s double series inequality and its integral analogue, involving a series of nonnegative terms. In addition, we obtain some new Hilbert type inequalities.

These inequalities improve the results obtained by Pachpatte [9].

2. Main results. Our main results are given in the following theorems.

Theorem2.1. Letp≥1,q≥1,0< α, and let{am}and{bn}be two nonnegative sequences of real numbers defined form=1,2, . . . , kandn=1,2, . . . , r ,wherek,rare the natural numbers and defineAm=m

s=1as,Bn=n

t=1bt. Then k

m=1

r n=1

A^pmBn^q

m^α+n^α1/α ≤C(p, q, k, r;α)



^k

m=1

(k−m+1)

A^p−1m am2





1/2

×



^r

n=1

(r−n+1) Bn^q−1bn

2





1/2

,

(2.1)

unless{am}or{bn}is null, where

C(p, q, k, r;α)= 1

2 1/α

pq

kr . (2.2)

Proof. By using the following inequality (see [1,7]), _n

m=1

zm

β

≤β n m=1

zm

_m

k=1

zk

β−1

, (2.3)

whereβ≥1 is a constant andzm≥0,(m=1,2, . . .), it is easy to observe that A^pm≤p

m s=1

asA^p−1s , m=1,2, . . . , k, Bn^q≤q n t=1

atB_t^q−1, n=1,2, . . . , r . (2.4) From (2.4) and using the Schwarz inequality and the elementary inequality

_n

i=1

ai

1/n

≤ _n

i=1

a^α_i n

1/α

, 0< α,(see [4]), (2.5) (forai,i=1,2, . . . , n, nonnegative real numbers) we observe that

A^pmBn^q≤pq _m

s=1

asA^p−1s

_n

t=1

atB_t^q−1

≤pq(m)^1/2 _m

s=1

asA^p−1s 2

1/2

(n)^1/2 _n

t=1

atB^q−1_t 2

1/2

≤pq

m^α+n^α 2

1/α_m

s=1

asA^p−1s 2

1/2_n

t=1

atB_t^q−12

1/2

.

(2.6)

Dividing both sides of (2.6) by(m^α+n^α)^1/α, and then taking the sum overnfrom 1 torfirst and then the sum overmfrom 1 tokand using the Schwarz inequality and then interchanging the order of the summations (see [7,8]) we observe that

k m=1

r n=1

A^pmB^qn

m^α+n^α1/α

≤pq 1

2

1/α_k

m=1

_m

s=1

asA^p−1s 2

1/2_r

n=1

_n

t=1

btB^q−1_t 2

1/2

≤pq 1

2 1/α

(k)^1/2 _k

m=1

_m

s=1

asA^p−1s 2

1/2

(r )^1/2 _r

n=1

_n

t=1

btB_t^q−12

1/2

=pq kr

1 2

1/α_k

s=1

asA^p−1s 2

_k

m=s

1

1/2_r

t=1

btB^q−1_t 2

_r

n=t

1 1/2

=C(p, q, k, r;α) _k

s=1

asA^ps⁻¹2

(k−s+1)

1/2_r

t=1

btB_t^q−12

(r−t+1) 1/2

=C(p, q, k, r;α) _k

m=1

(k−m+1)

amA^p−1m 2

1/2_r

n=1

(r−n+1)

bnBn^q−12

1/2

. (2.7) This completes the proof.

Remark2.2. InTheorem 2.1, settingα≡1, we haveTheorem 1.3. If we takep= q=1 inTheorem 2.1, then the inequality of the result ofTheorem 2.1reduces to the following inequality:

k m=1

r n=1

AmBn

m^α+n^α1/α≤C(1,1, k, r;α) _k

m=1

(k−m+1) am

2

1/2

× _r

n=1

(r−n+1) bn

2

1/2

,

(2.8)

whereC(1,1, k, r;α)is obtained by takingp=q=1 in (2.2).

Our next result deals with further generalization of the inequality obtained in Remark 2.2.

Theorem2.3. Let{am},{bn},Am, andBnbe as defined inTheorem 2.1. Let{pm} and{qn}be two nonnegative sequences form=1,2, . . . , kandn=1,2, . . . , r, and define Pm=m

s=1ps,Qn=n

t=1qt. Letφandψbe two real-valued, nonnegative, convex, and submultiplicative functions defined onR+=[0,∞). Then

k m=1

r n=1

φ Am

ψ Bn

m^α+n^α1/α ≤M(k, r;α) k

m=1

(k−m+1)

pmφ am

pm

²1/2

× _r

n=1

(r−n+1)

qnφ bn

qn

²1/2

,

(2.9)

where

M(k, r;α)=1 2

1/α _k

m=1

φ Pm Pm

21/2_r

n=1

ψ Qn Qn

21/2

. (2.10)

Proof. From the hypotheses ofφandψand by using Jensen’s inequality and the Schwarz inequality (see [5]), it is easy to observe that

φ Am

=φ Pm

m

s=1psas/ps

m s=1ps

≤φ Pm

φ

ms=1mpsas/ps s=1ps

≤φ Pm

Pm

m s=1

psφ as

ps

≤φ Pm

Pm

(m)^1/2 _m

s=1

psφ

as

ps

21/2

,

(2.11)

and similarly,

ψ Bn

≤ψ Qn Qn

(n)^1/2



ⁿ

t=1

qtψ

bt

qt

2



1/2

. (2.12)

From (2.11) and (2.12) and using the elementary inequality _n

i=1

ai

1/n

≤ _n

i=1

a^α_i n

1/α

, 0< α, (2.13)

(forai,i=1,2, . . . , n, nonnegative real numbers) we observe that

φ Am

ψ Bn

≤

m^α+n^α 2

1/α φ

Pm Pm

_m

s=1

psφ

as

ps

21/2

× ψ

Qn

Qn

_n

t=1

qtψ

bt

qt

21/2 .

(2.14)

Dividing both sides of the above inequality by(m^α+n^α)^1/α, and then taking the sum overnfrom 1 tor first and then the sum overmfrom 1 tokand using the Schwarz inequality and then interchanging the order of the summations we observe that

k m=1

r n=1

φ Am

ψ Bn

m^α+n^α1/α

≤ 1

2

1/α _k

m=1

φ Pm

Pm

_m

s=1

psφ

as

ps

21/2

× _r

n=1

ψ Qn

Qn

_n

t=1

qtψ

bt

qt

21/2

≤ 1

2

1/α _k

m=1

φ Pm

Pm

21/2 _k

m=1

_m

s=1

psφ

as

ps

21/2

× _r

n=1

ψ Qn Qn

21/2_r

n=1

_n

t=1

qtψ

bt

qt

21/2

=M(k, r;α) _k

s=1

psφ

as

ps

2_k

m=s

1

1/2_r

t=1

qtψ

bt

qt

2_r

n=t

1 1/2

=M(k, r;α) _k

s=1

psφ

as

ps

2

(k−s+1)

1/2_r

t=1

qtψ

bt

qt

2

(r−t+1) 1/2

=M(k, r;α) _k

m=1

(k−m+1)

pmφ am

pm

21/2_r

n=1

(r−n+1)

qnψ bn

qn

21/2

.

(2.15) The proof is complete.

Remark2.4. By applying the elementary inequality (see [4]) _n

i=1

ai

1/n

≤ _n

i=1

a^γ_i n

1/γ

, 0< γ, (2.16)

(forai,i=1,2, . . . , n, nonnegative real numbers) on the right sides of the inequalities

in Theorems2.1and2.3, we get, respectively, the following inequalities:

k m=1

r n=1

A^pmB^qn

m^α+n^α1/α

≤C1

_k

m=1

(k−m+1) A^pm⁻¹am

2

γ

+ _r

n=1

(r−n+1) B^qn⁻¹bm

2

γ1/γ

,

(2.17)

whereC1=(1/2)^1/γC(p, q, k, r;α), and k

m=1

r n=1

φ Am

ψ Bn

m^α+n^α1/α

≤M1

_k

m=1

(k−m+1)

pmφ am

pm

2γ

+ _r

n=1

(r−n+1)

qnψ bn

qn

2γ1/γ

,

(2.18) whereM1=(1/2)^1/γM(k, r;α), which we believe are new to the literature.

The following theorems deal with slight variants of (2.9) given inTheorem 2.3.

Theorem2.5. Let{am}and{bn}be as defined inTheorem 2.1, and defineAm= 1/mm

s=1as andBn=1/nn

t=1bt, form=1,2, . . . , kandn=1,2, . . . , r, wherek, r are the natural numbers. Let φandψ be two real-valued, nonnegative, and convex functions defined onR+=[0,∞). Then

k m=1

r n=1

mn

m^α+n^α1/αφ Am

ψ Bn

≤C(1,1, k, r;α) _k

m=1

(k−m+1) φ

am2

1/2

× r

n=1

(r−n+1) ψ

bn

2

1/2

,

(2.19) whereC(1,1, k, r )is defined by takingp=q=1in (2.2).

Proof. From the hypotheses and by using Jensen’s inequality and the Schwarz inequality, it is easy to observe that

φ Am

=φ 1

m m s=1

as

≤ 1 m

m s=1

φ as

≤ 1 m(m)^1/2

_m

s=1

φ as

2

1/2

,

ψ Bm

=ψ 1

n n t=1

bt

≤ 1 n

n t=1

ψ bt

≤ 1 n(n)^1/2

_n

t=1

ψ bt

2

1/2

.

(2.20)

The rest of the proof can be completed by following the same steps as in the proofs of Theorems2.1and2.3with suitable changes and hence we omit the details.

Theorem 2.6. Let {am}, {bn}, {pm}, {qn}, Pm, and Qn be as in Theorem 2.3, and define Am =1/Pmm

s=1psas and Bn = 1/Qnn

t=1qtbt, for m= 1,2, . . . , k and

n=1,2, . . . , r, where k, r are the natural numbers. Let φ and ψ be as defined in Theorem 2.5. Then

k m=1

r n=1

PmQnφ Am

ψ Bn

m^α+n^α1/α ≤C(1,1, k, r;α) _k

m=1

(k−m+1) psφ

as

2

1/2

× _r

n=1

(r−n+1) qtψ

bt

2

1/2

,

(2.21)

whereC(1,1, k, r;α)is defined by takingp=q=1in (2.2).

Proof. From the hypotheses and by using Jensen’s inequality and the Schwarz inequality, it is easy to observe that

φ Am

=φ 1

Pm

m s=1

psas

≤ 1 Pm

m s=1

psφ as

≤ 1 Pm

(m)^1/2 _m

s=1

ps φ

as2

1/2

,

φ Bn

=ψ 1

Qn

n t=1

qtbt

≤ 1 Qn

n t=1

qtψ bt

≤ 1 Qn

(n)^1/2 _n

t=1

qtψ bt

2

1/2

.

(2.22) The rest of the proof can be completed by following the same steps as in the proofs of Theorems2.1and2.3with suitable changes and hence we omit the details.

3. Integral analogues. In this section, we present the integral analogues of the in- equalities given in Theorems2.1,2.3,2.5, and2.6, which in fact are motivated by the integral analogue of Hilbert’s inequality given inTheorem 1.2.

An integral analogue ofTheorem 2.1is given in the following theorem.

Theorem3.1. Letp≥1,q≥1,0< α≤1andf (σ )≥0,g(τ)≥0forσ ∈(0, x), τ∈(0, y), wherex,yare positive real numbers, defineF (s)=s

0f (σ )dσ andG(t)= t

0g(τ)dτ, fors∈(0, x),t∈(0, y). Then x

0

y 0

F^p(s)G^q(t)

s^α+t^α1/αds dt≤D(p, q, x, y;α) x

0(x−s) Ff(s)2

ds 1/2

× y

0

(y−t) Gg(t)2

dt 1/2

,

(3.1)

unlessf≡0org≡0, whereFf(s)=F^p−1(s)f (s),Gg(t)=G^q−1(t)g(t), and D(p, q, x, y;α)=

1 2

1/α

pq

xy. (3.2)

Proof. From the hypotheses ofF (s)andG(t), it is easy to observe that F^p(s)=p

s

0F^p−1(σ )f (σ )dσ , s∈(0, x), G^q(t)=q

t

0G^q−1(τ)g(τ)dτ, t∈(0, y).

(3.3)

From (3.3) and using the Schwarz inequality and the elementary inequality _n

i=1

ai

1/n

≤ _n

i=1

a^α_i n

1/α

, 0< α, (3.4)

(forai,i=1,2, . . . , n, nonnegative real numbers) we observe that

F^p(s)G^q(t)=pq s

0

F^p−1(σ )f (σ )dσ t

0

G^q−1(τ)g(τ)dτ

≤pq(s)^1/2 s

0

F^p−1(σ )f (σ )2

dσ 1/2

(t)^1/2 t

0

G^q−1(τ)g(τ)2

dτ 1/2

≤pq

s^α+t^α 2

1/αs 0

F^p−1(σ )f (σ )2

dσ 1/2t

0

G^q−1(τ)g(τ)2

dτ 1/2

. (3.5) Dividing both sides of the above inequality by(s^α+t^α)^1/α, and then integrating over tfrom 0 toyfirst and then integrating the resulting inequality oversfrom 0 toxand using the Schwarz inequality we observe that

x 0

y 0

F^p(s)G^q(t) s^α+t^α1/αds dt

≤pq 1

2

1/αx 0

s 0

Ff(σ )2

dσ 1/2

dt y

0

t 0

Gg(τ)2

dτ 1/2

dt

≤pq 1

2 1/α

(xy)^1/2 x

0

s 0

Ff(σ )2

dσ

ds

1/2y 0

t 0

Gg(τ)2

dτ

dt 1/2

=D(p, q, x, y;α) x

0

(x−s) Ff(s)2

ds

1/2y 0

(y−t) Gg(t)2

dt 1/2

,

(3.6) whereFf(σ )=F^p−1(σ )f (σ ),Gg(τ)=G^q−1(τ)g(τ). This completes the proof.

Remark3.2. In the special case whenp=q=1, inequality (3.1) inTheorem 3.1 reduces to the following inequality:

x 0

y 0

F (s)G(t) s^α+t^α1/αds dt

=D(1,1, x, y;α) x

0

(x−s)f²(s)ds

1/2y 0

(y−t)g²(t)dt 1/2

,

(3.7)

whereD(1,1, x, y;α)is obtained by takingp=q=1 in (3.2).

The integral analogues of the inequalities in Theorems2.3,2.5, and2.6are estab- lished in the following theorems.

Theorem3.3. Letf,g,F,Gbe as inTheorem 3.1. Letp(σ )andq(τ)be two positive functions defined forσ ∈(0, x),τ∈(0, y), and defineP (s)=s

0p(σ )dσ andQ(t)= t

0q(τ)dτ, fors∈(0, x),t∈(0, y), wherex,yare positive real numbers. Letφand ψbe as inTheorem 2.3. Then

x 0

y 0

φ F p(s)

ψ G(t)

s^α+t^α1/α ds dt≤L(x, y;α) x

0

(x−s)

p(s)φ f (s)

p(s) ²

ds 1/2

× y

0(y−t)

q(t)ψ g(t)

q(t) 2

dt 1/2

,

(3.8)

where

L(x, y;α)=1 2

1/αx 0

φ P (s) P (s)

2

ds

1/2x 0

ψ Q(t) Q(t)

2

dt 1/2

. (3.9)

Proof. From the hypotheses and by using Jensen’s inequality and the Schwarz inequality, it is easy to observe that

φ F (s)

=φ







P (s) s

0

P (s)

f (σ )/p(σ ) dσ

s

0

p(σ )dσ







≤φ P (s) P (s)

s 0P (σ )φ

f (σ ) p(σ )

dσ

≤φ P (s) P (s) (s)^1/2

s 0

P (σ )φ

f (σ ) p(σ )

2

dσ 1/2

,

(3.10)

and similarly,

ψ G(t)

≤ψ Q(t) Q(t) (t)^1/2

t 0

q(τ)ψ

g(τ) q(τ)

2

dτ 1/2

. (3.11)

From (3.10) and (3.11) and using the elementary inequality _n

i=1

ai

1/n

≤ _n

i=1

a^α_i n

1/α

, 0< α, (3.12)

(forai,i=1,2, . . . , n, nonnegative real numbers) we observe that

φ F (s)

ψ G(t)

≤

s^α+t^α 2

1/α φ

P (s) P (s) (s)^1/2

s 0

P (σ )φ

f (σ ) p(σ )

2

dσ 1/2

× ψ

Q(t) Q(t) (t)^1/2

t 0

q(τ)ψ

g(τ) q(τ)

2

dτ 1/2

.

(3.13)

The rest of the proof can be completed by following the same steps as in the proof of Theorem 3.1and closely looking at the proof ofTheorem 2.3, and hence we omit the details.

Theorem 3.4. Letf, g, be as in Theorem 3.1, and define F (s)=s

0f (σ )dσ and G(t)=t

0g(τ)dτ, fors∈(0, x),t∈(0, y), wherex,yare positive real numbers. Let φandψbe as inTheorem 2.5. Then

x 0

y 0

st

s^α+t^α1/αφ F (s)

ψ G(t)

ds dt

≤D(1,1, x, y;α) x

0(x−s) φ

f (σ )2

ds

1/2y 0(y−t)

ψ g(t)2

dt 1/2

, (3.14) whereD(1,1, x, y;α)is obtained by takingp=q=1in (3.2).

Theorem 3.5. Letf,g,p, q, P, andQbe as in Theorem 3.3, and defineF (s)= 1/P (s)s

0p(σ )f (σ )dσ andG(t)=1/Q(t)t

0q(τ)g(τ)dτ, for s∈(0, x), t∈(0, y), wherex,yare positive real numbers. Letφandψbe as inTheorem 2.5. Then

x 0

y 0

P (s)Q(s)φ F (s)

ψ G(t) s^α+t^α1/α ds dt

≤D(1,1, x, y;α) x

0(x−s) p(s)φ

f (s)2

ds 1/2

× y

0

(y−t) q(t)ψ

g(t)2

dt 1/2

,

(3.15)

whereD(1,1, x, y;α)is obtained by takingp=q=1in (3.2).

The proofs of Theorems3.4and3.5can be completed by following the proof of Theorem 3.3and by closely looking at the proofs of Theorems 2.5and 2.6 and by making use of the integral versions of Jensen’s and the Schwarz inequalities. Here, we omit the details.

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Young-Ho Kim: Department of Applied Mathematics (or, Brain Korea 21 Project Corps), Changwon National University, Changwon641-773, Korea

E-mail address:[email protected]

Journal of Applied Mathematics and Decision Sciences

Special Issue on

Intelligent Computational Methods for Financial Engineering

Call for Papers

As a multidisciplinary field, financial engineering is becom- ing increasingly important in today’s economic and financial world, especially in areas such as portfolio management, as- set valuation and prediction, fraud detection, and credit risk management. For example, in a credit risk context, the re- cently approved Basel II guidelines advise financial institu- tions to build comprehensible credit risk models in order to optimize their capital allocation policy. Computational methods are being intensively studied and applied to im- prove the quality of the financial decisions that need to be made. Until now, computational methods and models are central to the analysis of economic and financial decisions.

However, more and more researchers have found that the financial environment is not ruled by mathematical distribu- tions or statistical models. In such situations, some attempts have also been made to develop financial engineering mod- els using intelligent computing approaches. For example, an artificial neural network (ANN) is a nonparametric estima- tion technique which does not make any distributional as- sumptions regarding the underlying asset. Instead, ANN ap- proach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting pa- rameters to obtain the desired results. The main aim of this special issue is not to merely illustrate the superior perfor- mance of a new intelligent computational method, but also to demonstrate how it can be used e

ectively in a financial engineering environment to improve and facilitate financial decision making. In this sense, the submissions should es- pecially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelli- gent, easy-to-use, and/or comprehensible computational sys- tems (e.g., decision support systems, agent-based system, and web-based systems)

This special issue will include (but not be limited to) the following topics:

• Computational methods

: artificial intelligence, neu- ral networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learn- ing, multiagent learning

• Application fields

: asset valuation and prediction, as- set allocation and portfolio selection, bankruptcy pre- diction, fraud detection, credit risk management

• Implementation aspects

: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, imple- mentation

Authors should follow the Journal of Applied Mathemat- ics and Decision Sciences manuscript format described at the journal site

Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Track- ing System at

lowing timetable:

Manuscript Due December 1, 2008 First Round of Reviews March 1, 2009 Publication Date June 1, 2009

Guest Editors

Lean Yu,

Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China;

Department of Management Sciences, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong;

[email protected]

Shouyang Wang,

Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China; [email protected]

K. K. Lai,

Department of Management Sciences, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong; [email protected]

Hindawi Publishing Corporation http://www.hindawi.com