On Some Extensions of Hardy-Hilbert’s Inequality and Applications

Volume 2008, Article ID 546829,14pages doi:10.1155/2008/546829

Research Article

On Some Extensions of Hardy-Hilbert’s Inequality and Applications

Laith Emil Azar

Department of Mathematics, Al Al-Bayt University, P.O. Box. 130095, Mafraq 25113, Jordan

Correspondence should be addressed to Laith Emil Azar,azar [email protected] Received 23 October 2007; Accepted 2 January 2008

Recommended by Shusen Ding

By introducing some parameters we establish an extension of Hardy-Hilbert’s integral inequality and the corresponding inequality for series. As an application, the reverses, some particular results and their equivalent forms are considered.

Copyrightq2008 Laith Emil Azar. 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 Iffx,gx≥0, 0<_∞

0 f²xdx <∞, and 0<_∞

0 g²xdx <∞, thensee1 _∞

0

fxgy

xy dx dy < π

∞ 0

f²xdx

1/2 ∞ 0

g²xdx 1/2

, 1.1

_∞

0

fxgy

max{x, y}dx dy <4

∞ 0

f²xdx1/2 ∞ 0

g²xdx1/2

, 1.2

where the constant factorsπ and 4 are the best possible in1.1and1.2, respectively. In- equality 1.1 is called Hilbert’s integral inequality and1.2 is called Hilbert’s type which have been extended by Hardysee2as follows: ifp > 1, 1/p1/q 1,fx,gx > 0, 0<_∞

0 f^pxdx <∞, and 0<_∞

0 g^qxdx <∞, then _∞

0

fxgy

xy dx dy < π sinπ/p

∞ 0

f^pxdx

1/p ∞ 0

g^qxdx 1/q

, 1.3

_∞

0

fxgy

max{x, y}dx dy < pq

∞ 0

f^pxdx

1/p ∞ 0

g^qxdx 1/q

, 1.4

where the constant factorsπ/sinπ/pandpqare the best possible in1.3and1.4, respec- tively. Hardy-Hilbert’s inequality and its applications are important in analysissee3. Re- cently, Yang4gave some generalizations and the reverse form of1.3as follows: ifp > 1, 1/p1/q 1,r >1, 1/r1/s1,λ > 0,fx, gx ≥0, 0<_∞

0 x^{p1−λ/r−1}f^pxdx <∞, and 0<_∞

0 x^{q1−λ/s−1}g^qxdx <∞, then _∞

0

fxgy

x^λy^λ dx dy < π λsinπ/r

∞ 0

x^{p1−λ/r−1}f^pxdx

1/p ∞ 0

x^{q1−λ/s−1}g^qxdx 1/q

, 1.5

where the constant factorπ/λsinπ/ris the best possible.

The corresponding inequalities for series1.3and1.4are

∞ n1

∞ m1

a_mb_n

mn < π sinπ/p

_∞

n0

a^p_n

1/p_∞

n0

b^q_n 1/q

,

∞ n1

∞ m1

ambn

max{m, n} < pq ∞

n0

a^p_n

1/p∞ n0

b^q_n 1/q

,

1.6

where the sequences{an}and{bn}are such that 0 <_∞

n0a^p_n <∞, 0 <_∞

n0a^q_n <∞,and the constant factorπ/sinπ/pandpqare the best possible. By introducing a parameter 0< λ≤2, some extensions of1.6 pq2were given by Yang5,6as

∞ n1

∞ m1

ambn

m^λn^λ <π λ

∞ n0

n^1−λa²_n

1/2∞ n0

n^1−λb_n² 1/2

,

∞ n1

∞ m1

a_mb_n max

m^λ, n^λ < 4 λ

_∞

n0

n^1−λa²_n

1/2_∞

n0

n^1−λb²_n 1/2

.

1.7

Very recently, in7the following extensions were given:

_∞

0

fxgx

Amin{x, y}Bmax{x, y}dx dy < DA, B ∞

0

f²xdx

_1/2∞ 0

g²xdx _1/2

,

∞ n1

∞ m1

ambn

Amin{m, n}Bmax{m, n} < DA, B∞ n1

a²_n

1/2∞ n1

b_n² 1/2

,

1.8

where the constant factorDA, B see7, Lemma 2.1is the best possible in both inequalities.

For more information related to this subject see, for example,8,9.

In this paper by introducing some parameters, we generalize1.8and we obtain the reverse form for each of them. Some particular results and the equivalent form are also consid- ered.

2. Main results

Lemma 2.1. Suppose that λ > 0, A ≥ 0, B > 0. Define the weight coefficients ωλA, B, x and ωλA, B, yby

ωλA, B, x: _∞

0

x^λ/2y^−1λ/2 Amin

x^λ, y^λ Bmax

x^λ, y^λ dy, 2.1

ω_λA, B, y: _∞

0

x^−1λ/2y^λ/2 Amin

x^λ, y^λ Bmax

x^λ, y^λ dx, 2.2

thenω_λA, B, x ω_λA, B, y C_λA, Bis a constant defined by

C_λA, B

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ 4 λ√

ABarctan A

B forA >0, B >0, 4

λB forA0, B >0.

2.3

Proof. Settingt y/x^λ, we get

ωλA, B, x _∞

0

x^λ/2y^−1λ/2 Amin

x^λ, y^λ Bmax

x^λ, y^λ dy, 1

λ _∞

0

t^−1/2

Amin{1, t}Bmax{1, t}dt:I,

2.4

iforA,B >0, we obtain

I 1 λ

1 0

t^−1/2 AtBdt

_∞

1

t^−1/2 ABtdt

1 λ

2

√AB √

A/B 0

dt t²1 2

√AB √

A/B 0

dt t²1

4 λ√

ABarctan

A B;

2.5

ii forA0,B >0, we find

I 1 λ

1 0

t^−1/2 B dt

_∞

1

t^−1/2 Bt dt

4

λB. 2.6

Hence,ω_λA, B, x C_λA, B. By the symmetry we still haveω_λA, B, y C_λA, B.

The lemma is proved.

Lemma 2.2. Forp >1or 0< p <1, 1/p1/q1,λ >0,A≥0,B >0 and 0< ε < pλ/2, setting

Jε _∞

1

x^{λ/2−1−ε/p}y^{λ/2−1−ε/q} Amin

x^λ, y^λ Bmax

x^λ, y^λ dx dy, 2.7 then forε→0,

1 ε

C_λA, B o1

−O1< Jε<1 ε

C_λA, B o1

. 2.8

Proof. Settingt x/y^λ, we find

Jε _∞

1

x^{λ/2−1−ε/p}y^{λ/2−1−ε/q} Amin

x^λ, y^λ Bmax

x^λ, y^λ dx dy 1

λ _∞

1

y^−1−ε _∞

y^−λ

t^{−1/2−ε/λp}

Amin{t,1}Bmax{t,1}dt dy 1

λε _∞

0

t^{−1/2−ε/λp}

Amin{t,1}Bmax{t,1}dt dy

−1 λ

_∞

1

y^−1−ε _y−λ

0

t^{−1/2−ε/λp}

Amin{t,1}Bmax{t,1}dt dy

1 ε

CλA, B o1

−1 λ

_∞

1

y^−1−ε _y^−λ

0

t^{−1/2−ε/λp} AtB dt dy

≥ 1 ε

CλA, B o1

−1 λ

_∞

1

y⁻¹ _y^−λ

0

t^{−1/2−ε/λp} B dt dy 1

ε

C_λA, B o1

−O1.

2.9

On the other hand,

Jε _∞

1

x^{λ/2−1−ε/p}y^{λ/2−1−ε/q} Amin

x^λ, y^λ Bmax

x^λ, y^λ dx dy

<

_∞

1

∞ 0

x^{λ/2−1−ε/p} Amin

x^λ, y^λ Bmax

x^λ, y^λ dx

y^{λ/2−1−ε/q}dy

1 ε

CλA, B o1 .

2.10

Hence,2.8is valid. The lemma is proved.

Theorem 2.3. If p > 1, 1/p1/q 1,λ > 0, A ≥ 0,B > 0, fx,gx ≥ 0 such that 0 <

_∞

0 x^{p1−λ/2−1}f^pxdx <∞, 0<_∞

0 x^{q1−λ/2−1}g^qxdx <∞, then

S: _∞

0

fxgx Amin

x^λ, y^λ Bmax

x^λ, y^λ dx dy

< CλA, B ^∞

0

x^{p1−λ/2−1}f^pxdx1/p ∞ 0

x^{q1−λ/2−1}g^qxdx1/q

,

2.11

where the constant factorC_λA, Bdefined in2.3is the best possible. In particular,

iforλAB1,C₁1,1 π, and inequality2.11reduces to Hardy-Hilbert’s inequality _∞

0

fxgx

xy dx dy < π

∞ 0

x^p/2−1f^pxdx

1/p ∞ 0

x^q/2−1g^qxdx 1/q

; 2.12

ii forA0,λB1,C10,1 4 and2.11reduces to Hardy-Hilbert’s-type inequality _∞

0

fxgx

max{x, y}dx dy <4

∞ 0

x^p/2−1f^pxdx

1/p ∞ 0

x^q/2−1g^qxdx 1/q

. 2.13

Proof. By the Holder inequality, taking into account2.1, we get

S _∞

0

1 Amin

x^λ, y^λ Bmax x^λ, y^λ

_1/p

x^1−λ/2/q y^1−λ/2/pfx

×

1 Amin

x^λ, y^λ}Bmax x^λ, y^λ}

_1/qy^1−λ/2/p x^1−λ/2/qgy

dx dy

≤ ^∞

0

x^{1−λ/2p−1}y^λ/2−1 Amin

x^λ, y^λ}Bmax

x^λ, y^λ}f^pxdx^1/p

× ^∞

0

y^{1−λ/2q−1}x^λ/2−1 Amin

x^λ, y^λ Bmax

x^λ, y^λ g^qydy^1/q

^∞

0

ω_λA, B, xx^{p1−λ/2−1}f^pxdx

1/p ∞ 0

ω_λA, B, yy^{q1−λ/2−1}g^qydy 1/q

≤C_λA, B ^∞

0

x^{p1−λ/2−1}f^pxdx1/p ∞ 0

y^{q1−λ/2−1}g^qydy1/q

.

2.14

If2.14takes the form of equality, then there exist constantsMandNwhich are not all zero such that

M x^{1−λ/2p−1}y^λ/2−1

Amin

x^λ, y^λ Bmax

x^λ, y^λ f^px N y^{1−λ/2q−1}x^λ/2−1 Amin

x^λ, y^λ Bmax

x^λ, y^λ g^qy, Mx^p1−λ/2f^px Ny^q1−λ/2g^qy, a.e. in0,∞×0,∞

2.15

Hence, there exists a constantcsuch that

Mx^p1−λ/2f^px Ny^q1−λ/2g^qy c a.e. in0,∞. 2.16

We claim thatM0. In fact, ifM /0, then

x^{p1−λ/2−1}f^px c

Mx a.e. in0,∞ 2.17

which contradicts the fact that 0<_∞

0 x^{p1−λ/2−1}f^pxdx <∞. Hence, by2.14we get2.11.

If the constant factorC_λA, Bis not the best possible, then there exists a positive constant KwithK < CλA, B, thus2.11is still valid if we replaceCλA, BbyK. For 0< ε < pλ/2, setting fand g asfx gx 0 forx ∈ 0,1,fx x^{λ/2−1−ε/p}; gx x^{λ/2−1−ε/q} for x∈1,∞, then we have

K

∞ 0

x^{p1−λ/2−1}f^pxdx1/p ∞ 0

x^{q1−λ/2−1}g^qxdx1/q

K

∞ 0

x^−1−εdx

1/p ∞ 0

x^−1−εdx 1/q

K ε.

2.18

By using2.8, we find _∞

0

fx gxdx dy Amin

x^λ, y^λ Bmax

x^λ, y^λ _∞

1

∞ 1

x^{λ/2−1−ε/p}dx Amin

x^λ, y^λ Bmax x^λ, y^λ

y^{λ/2−1−ε/q}dy

> 1 ε

C_λA, B o1

−O1.

2.19

Therefore, we get

1 ε

CλA, B o1

−O1< K

ε 2.20

or

1 λ

C_λA, B o1

−εO1< K. 2.21

Forε→ 0, it follows thatC_λA, B ≤ Kwhich contradicts the fact thatK < C_λA, B.

Hence, the constant factorCλA, Bin2.11is the best possible. The theorem is proved.

Theorem 2.4. If 0 < p < 1, 1/p1/q 1,λ > 0, A ≥ 0, B > 0, fx, gx ≥ 0 such that 0<_∞

0 x^{p1−λ/2−1}f^pxdx <∞, 0<0<_∞

0 x^{q1−λ/2−1}g^qxdx <∞, then _∞

0

fxgx Amin

x^λ, y^λ Bmax

x^λ, y^λ dx dy

> CλA, B ^∞

0

x^{p1−λ/2−1}f^pxdx1/p ∞ 0

x^{q1−λ/2−1}g^qxdx1/q

,

2.22

where the constant factorC_λA, Bdefined in2.3is the best possible. In particular,

iforλAB1,C₁1,1 π, and inequality2.22reduces to Hardy-Hilbert’s inequality _∞

0

fxgx

xy dx dy > π

∞ 0

x^p/2−1f^pxdx

_{1/p ∞}

0

x^q/2−1g^qxdx _1/q

, 2.23

ii forA0,λB1,C10,1 4 and2.22reduces to Hardy-Hilbert’s-type inequality _∞

0

fxgx

max{x, y}dx dy >4

∞ 0

x^p/2−1f^pxdx

_1/p ∞ 0

x^q/2−1g^qxdx _1/q

. 2.24

Proof. By reverse Holder’s inequality, and the same way, we have2.22. If the constant factor C_λA, Bin2.22is not the best possible, then there exists a positive constantH withH >

CλA, Bsuch that2.22is still valid if we replaceCλA, BbyH. For 0< ε < pλ/2, settingf andgas inTheorem 2.3, then we have

H

∞ 0

x^{p1−λ/2−1}f^pxdx1/p ∞ 0

x^{q1−λ/2−1}g^qxdx1/q

H

∞ 0

x^−1−εdx

1/p ∞ 0

x^−1−εdx 1/q

H ε .

2.25

By using2.8, we find _∞

0

fx gxdx dy Amin

x^λ, y^λ Bmax

x^λ, y^λ _∞

1

∞ 1

x^{λ/2−1−ε/p}dx Amin

x^λ, y^λ Bmax x^λ, y^λ

y^{λ/2−1−ε/q}dy

< 1 ε

C_λA, B o1 .

2.26

Therefore, we get

1 ε

CλA, B o1

> H

ε, 2.27

or

C_λA, B o1≥H. 2.28 Forε→0, it follows thatCλA, B≥Hwhich contradicts the fact thatH > CλA, B. Hence, the constant factorCλA, Bin2.22is the best possible. The theorem is proved.

Theorem 2.5. Under the assumption ofTheorem 2.3, _∞

0

y^λp/2−1

∞ 0

fx Amin

x^λ, y^λ Bmax

x^λ, y^λ dxp

dy <

C_λA, B_p∞

0

x^{p1−λ/2−1}f^pxdx, 2.29 where the constant factorCλA, B^pis the best possible. Inequalities2.11and2.29are equivalent.

Proof. Setting

gy y^λp/2−1

∞ 0

fx Amin

x^λ, y^λ Bmax

x^λ, y^λ dx p−1

, 2.30

then by2.11we have _∞

0

y^{q1−λ/2−1}g^qydy _∞

0

y^λp/2−1

∞ 0

fx Amin

x^λ, y^λ Bmax

x^λ, y^λ dx p

dy

_∞

0

∞ 0

fx Amin

x^λ, y^λ Bmax

x^λ, y^λ dx

×

y^λp/2−1

∞ 0

fx Amin

x^λ, y^λ Bmax

x^λ, y^λ dx p−1

dy

_∞

0

fxgy

Amin

x^λ, y^λ Bmax

x^λ, y^λ dx dy

≤C_λA, B ^∞

0

x^{p1−λ/2−1}f^pxdx1/p ∞ 0

y^{q1−λ/2−1}g^qydy1/q

. 2.31 Hence, we obtain

_∞

0

y^{q1−λ/2−1}g^qydy≤

CλA, Bp_∞

0

x^{p1−λ/2−1}f^pxdx. 2.32

Thus, by2.11, both2.31and2.32keep the form of strict inequalities, then we have2.29.

By Holder’s inequality, we find _∞

0

fxgy Amin

x^λ, y^λ Bmax

x^λ, y^λ dx dy

_∞

0

y^λ/2−1/p

_∞

0

fx Amin

x^λ, y^λ Bmax

x^λ, y^λ dx

y^1/p−λ/2gy dy

≤ ^∞

0

y^pλ/2−1

∞ 0

fx Amin

x^λ, y^λ Bmax

x^λ, y^λ dx

p_1/p ∞ 0

y^{q1−λ/2−1}g^qydy _1/q

. 2.33 Therefore, by2.29we have2.11, and inequalities2.29and2.11are equivalent. If the constant factor in2.29is not the best possible, then by2.33we can get a contradiction that the constant factor in2.11is not the best possible. The theorem is proved.

Theorem 2.6. Under the assumption ofTheorem 2.4, _∞

0

y^λp/2−1

∞ 0

fx Amin

x^λ, y^λ Bmax

x^λ, y^λ dx p

dy >

C_λA, Bp_∞

0

x^{p1−λ/2−1}f^pxdx, 2.34 where the constant factorCλA, B^pis the best possible. Inequalities2.22and2.34are equivalent.

The proof ofTheorem 2.6is similar to that ofTheorem 2.5, so we omit it.

3. Discrete analogous

Lemma 3.1. Suppose that 0 < λ ≤ 2,A≥ 0,B > 0. Then the weight coefficientsλA, B, mand _λA, B, n,defined, respectively, by

λA, B, m:^∞

n1

m^λ/2n^−1λ/2 Amin

m^λ, n^λ Bmax

m^λ, n^λ m∈N, 3.1

_λA, B, n:^∞

m1

n^λ/2m^−1λ/2 Amin

m^λ, n^λ Bmax

m^λ, n^λ n∈N, 3.2

satisfy the following inequalities:

CλA, B

1−θλA, B, m

< λA, B, m< CλA, B, 3.3 CλA, B

1−θλA, B, n

< λA, B, n< CλA, B, 3.4

whereθ_λA, B, r: 1/C1A, B_r^−λ

0 t^−1/2/AtBdtO1/r^λ/2∈0,1 r ∈N r → ∞,

andCλA, Bis defined by2.3.

Proof. Since 0< λ≤2,A≥0,B >0, byLemma 2.1we get

λA, B, m<

_∞

0

m^λ/2y^−1λ/2 Amin

m^λ, y^λ Bmax

m^λ, y^λ dy, ωλA, B, m CλA, B.

3.5

On the other hand, we have

λA, B, m>

_∞

1

m^λ/2y^−1λ/2 Amin

m^λ, y^λ Bmax

m^λ, y^λ dy 1

λ _∞

m^−λ

t^−1/2

Amin{1, t}Bmax{1, t}dt I−1

λ _m^−λ

0

t^−1/2 AtBdt I

1−θ_λA, B, m ,

3.6

whereI 1/λC1A, Band

0< θ_λA, B, m 1 C1A, B

_m^−λ

0

t^−1/2

AtBdt <1. 3.7

Since

_m^−λ

0

t^−1/2 AtBdt≤

_m^−λ

0

t^−1/2

B dt 2

Bm^λ/2, 3.8

thenθ_λA, B, m O1/m^λ/2. Therefore,3.3is valid. By the symmetry,3.4is still valid. The lemma is proved.

Lemma 3.2. Ifp >0p /1, 1/p1/q1, 0< λ≤2,A≥0,B >0, and 0< ε < pλ/2,setting

Lε ^∞

n1

∞ m1

m^{λ/2−1−ε/p}n^{λ/2−1−ε/q} Amin

m^λ, n^λ Bmax

m^λ, n^λ , 3.9

then forε→0,

CλA, B−o1^∞

n1

1

n^1ε < Lε<

CλA, B o1 ^∞

n1

1

n^1ε. 3.10

Proof. Settingt x/n^λin the following, by3.4, we have

Lε<

∞ n1

_∞

0

x^{λ/2−1−ε/p}n^{λ/2−1−ε/q} Amin

x^λ, n^λ Bmax

x^λ, n^λ dx ^∞

n1

1 n^1ε

1 λ

_∞

0

t^{−1/2−ε/λp}

Amin{t,1}Bmax{t,1}dt

CλA, B o1 ^∞

n1

1 n^1ε

ε−→0 ,

Lε>

∞ n1

_∞

1

x^{λ/2−1−ε/p}n^{λ/2−1−ε/q} Amin

x^λ, n^λ Bmax

x^λ, n^λ dx ^∞

n1

1 n^1ε

1 λ

_∞

n^−λ

t^{−1/2−ε/λp}

Amin{t,1}Bmax{t,1}dt

^∞

n1

1 n^1ε

C_λA, B o1 −1 λ

_n^−λ

0

t^{−1/2−ε/λp} AtB dt

>

∞ n1

1 n^1ε

C_λA, B o1

−1 λ

∞ n1

1 n

_n^−λ

0

t^{−1/2−ε/λp}

B dt

^∞

n1

1 n^1ε

C_λA, B o1

− 1

λB/2−εB/p ∞ n1

1 n^1λ/2−ε/p

^∞

n1

1 n^1ε

C_λA, B o1 − 1 λB/2−εB/p

∞ n1

1 n^1λ/2−ε/p

∞ n1

1 n^1ε

−1

^∞

n1

1 n^1ε

C_λA, B−o1 ε−→0 .

3.11

Thus, inequality3.10holds. The lemma is proved.

Theorem 3.3. If p > 1, 1/p1/q 1, 0 < λ ≤ 2,A ≥ 0, B > 0,a_n, b_n ≥ 0 such that 0 <

_∞

n1n^{p1−λ/2−1}a^p_n<∞, 0<_∞

n1n^{q1−λ/2−1}b^q_n<∞, then

D:^∞

n1

∞ m1

a_mb_n Amin

m^λ, n^λ Bmax m^λ, n^λ

< CλA, B _∞

n1

n^{p1−λ/2−1}a^p_n

_1/p∞

n1

n^{q1−λ/2−1}b^q_{n 1/q}

,

3.12

where the constant factorCλA, Bdefined in2.3is the best possible. In particular,

iforλAB1,C₁1,1 π, and inequality3.12reduces to Hardy-Hilbert’s inequality ∞

n1

∞ m1

ambn

mn< π ∞

n1

n^p/2−1a^p_n

_1/p∞ n1

n^q/2−1b^q_{n 1/q}

; 3.13

ii forA0,λB1,C10,1 4 and3.12reduces to Hardy-Hilbert’s-type inequality ∞

n1

∞ m1

ambn

max{m, n} <4 ∞

n1

n^p/2−1a^p_n

1/p∞ n1

n^q/2−1b^q_n 1/q

. 3.14

Proof. By the Holder inequality, taking into account3.1, we get

D^∞

n1

∞ m1

1 Amin

m^λ, n^λ Bmax m^λ, n^λ

_1/p

m^1−λ/2/q n^1−λ/2/pa_m

×

1 Amin

m^λ, n^λ Bmax m^λ, n^λ

1/q

n^1−λ/2/p m^1−λ/2/qbn

≤ ∞

n1

∞ m1

1 Amin

m^λ, n^λ Bmax m^λ, n^λ

m^{1−λ/2p−1} n^1−λ/2 a^p_m

1/p

× _∞

n1

∞ m1

1 Amin

m^λ, n^λ Bmax m^λ, n^λ

n^{1−λ/2q−1} m^1−λ/2 b_n^q

_1/q

∞

m1

λA, B, mm^{p1−λ/2−1}a^p_m

1/p∞ n1

λA, B, nn^{q1−λ/2−1}b^q_n 1/q

.

3.15

Then, by3.3and3.4we obtain3.12.

It remains to show that the constant factorC_λA, Bis the best possible, to do that we set for 0< ε < pλ/2,a_mm^{λ/2−1−ε/p};b_nn^{λ/2−1−ε/q}, by3.9we have

∞ n1

∞ m1

a_mb_n Amin

m^λ, n^λ Bmax

m^λ, n^λ Lε. 3.16

If there exists a constant 0< K≤CλA, Bsuch that3.12is still valid if we replaceCλA, B byK, then in particular by3.10we find

C_λA, B−o1^∞

n1

1

n^1ε < Lε< K ∞

n1

n^{p1−λ/2−1}a^p_n

1/p∞ n1

n^{q1−λ/2−1}b_n^q 1/q

K ∞ n1

1 n^1ε,

3.17

it follows thatC_λA, B−o1< Kand thenC_λA, B≤Kε→0. Therefor,KC_λA, Bis the best constant factor in3.12. The theorem is proved.

Theorem 3.4. If 0 < p < 1, 1/p1/q 1, 0 < λ ≤ 2,A ≥ 0, B > 0,an,bn ≥ 0 such that 0<_∞

n1n^{p1−λ/2−1}a^p_n<∞, 0<_∞

n1n^{q1−λ/2−1}b_n^q<∞, then ∞

n1

∞ m1

ambn

Amin

m^λ, n^λ Bmax m^λ, n^λ

> C_λA, B∞ n1

1−θ_λA, B, n

n^{p1−λ/2−1}a^p_n

1/p∞ n1

n^{q1−λ/2−1}b_n^q 1/q

,

3.18

where the constant factorC_λA, Bdefined in2.4is the best possible. In particular, iforλAB1,C₁1,1 π, and

∞ n1

∞ m1

a_mb_n mn> π

∞ n1

1− 2

πarctan 1 n^1/2

n^p/2−1a^p_n

1/p∞ n1

n^q/2−1b^q_n 1/q

; 3.19

ii forA0,λB1,C₁0,1 4, and ∞

n1

∞ m1

a_mb_n max{m, n}>4

_∞

n1

1− 1

2n^1/2

n^p/2−1a^p_n

_1/p∞

n1

n^q/2−1b_n^q _1/q

. 3.20

Proof. By reverse Holder’s inequality, we get D^∞

n1

∞ m1

ambn

Amin

m^λ, n^λ Bmax m^λ, n^{λ ∞}

n1

∞ m1

1 Amin

m^λ, n^λ Bmax m^λ, n^λ

_1/p

m^1−λ/2/q n^1−λ/2/pa_m

×

1 Amin

m^λ, n^λ Bmax m^λ, n^λ

1/q

n^1−λ/2/p m^1−λ/2/qbn

≥∞ m1

λA, B, mm^{p1−λ/2−1}a^pm

1/p∞ n1

λA, B, nn^{q1−λ/2−1}b^qn

1/q

.

3.21

Then by3.3and3.4, in view ofq <0, we have3.18. For 0< ε < pλ/2, settingam m^{λ/2−1−ε/p},bnn^{λ/2−1−ε/q}m, n∈N. If there exists a constantK≥CλA, Bsuch that3.18is still valid if we replaceCλA, BbyK, then in particular by3.9and3.10we find

CλA, B o1 ^∞

n1

1

n^1ε > Lε

> K ∞

n1

1−θ_λA, B, n

n^{p1−λ/2−1}a^p_n

1/p∞ n1

n^{q1−λ/2−1}b^q_n 1/q

K _∞

n1

1 n^1ε−^∞

n1

O

1 n^λ/2

1 n^1ε

_1/p∞

n1

1 n^1ε

_1/q

K ∞ n1

1 n^1ε

1−∞

n1

1 n^1ε

−1∞ n1

O

1 n^λ/2

1 n^1ε

^1/p ,

3.22

it follows that

CλA, B o1 > K

1− ∞

n1

1 n^1ε

−1∞ n1

O

1 n^λ/2

1 n^1ε

_1/p

. 3.23

Hence, ifε → 0, we get CλA, B ≥ K. Thus,K CλA, Bis the best constant factor in 3.18.

Theorem 3.5. Under the assumption ofTheorem 3.3, ∞

n1

n^λp/2−1 _∞

m1

a_m Amin

m^λ, n^λ Bmax m^λ, n^λ

p

<

CλA, Bp∞ m1

m^{p1−λ/2−1}a^p_m, 3.24 where the constant factorCλA, B^pis the best possible. Inequalities3.12and3.24are equivalent.

Proof. Setting

bnn^λp/2−1 ∞

m1

am

Amin

m^λ, n^λ Bmax m^λ, n^λ

p−1

, 3.25

we get

∞ n1

n^{q1−λ/2−1}b_n^q∞

n1

∞ m1

ambn

Amin

m^λ, n^λ Bmax

m^λ, n^λ . 3.26 By3.12and using the same method ofTheorem 2.5, we obtain3.24. We may show that the constant factor in3.24is the best possible and inequality3.12is equivalent to3.24.

Theorem 3.6. Under the assumption ofTheorem 3.4, ∞

n1

n^λp/2−1 _∞

m1

a_m Amin

m^λ, n^λ Bmax m^λ, n^λ

p

>

C_λA, Bp∞ m1

m^{p1−λ/2−1}a^p_m, 3.27 where the constant factorCλA, B^pis the best possible. Inequalities3.18and3.27are equivalent.

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