This paper deals with some extensions of Hardy-Hilbert’s inequality with the best constant factors by introducing two parametersλandαand using the Beta function

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Volume 6, Issue 3, Article 81, 2005

ON BEST EXTENSIONS OF HARDY-HILBERT’S INEQUALITY WITH TWO PARAMETERS

BICHENG YANG DEPARTMENT OFMATHEMATICS

GUANGDONGINSTITUTE OFEDUCATION

GUANGZHOU, GUANGDONG510303 PEOPLE’SREPUBLIC OFCHINA

bcyang@pub.guangzhou.gd.cn

Received 23 February, 2005; accepted 17 June, 2005 Communicated by W.S. Cheung

ABSTRACT. This paper deals with some extensions of Hardy-Hilbert’s inequality with the best constant factors by introducing two parametersλandαand using the Beta function. The equiv- alent form and some reversions are considered.

Key words and phrases: Hardy-Hilbert’s inequality; Beta function; Hölder’s inequality.

2000 Mathematics Subject Classification. 26D15.

1. INTRODUCTION

Ifa_n,b_n≥0satisfy

0<

∞

X

n=1

a²_n<∞ and 0<

∞

X

n=1

b²_n <∞, then one has two equivalent inequalities as:

(1.1)

∞

X

n=1

∞

X

m=1

a_mb_n m+n < π

( _∞ X

n=1

a²_n

∞

X

n=1

b²_n )¹₂

and (1.2)

∞

X

n=1

∞

X

m=1

a_m m+n

!2

< π²

∞

X

n=1

a²_n,

where the constant factors π and π² are the best possible. Inequality (1.1) is well known as Hilbert’s inequality (cf. Hardy et al. [1]). In 1925, Hardy [2] gave some extensions of (1.1) and

ISSN (electronic): 1443-5756

c 2005 Victoria University. All rights reserved.

055-05

(1.2) by introducing the(p, q)−parameter as: Ifp >1, ¹_p + ¹_q = 1, a_n,b_n ≥0satisfy 0<

∞

X

n=1

a^p_n<∞ and 0<

∞

X

n=1

b^q_n <∞, then one has the following two equivalent inequalities:

(1.3)

∞

X

n=1

∞

X

m=1

a_mb_n

m+n < π sin

π p

( _∞

X

n=1

a^p_n

)¹_p ( _∞ X

n=1

b^q_n )¹_q

and (1.4)

∞

X

n=1

∞

X

m=1

a_m m+n

!p

<



 π sin

π p





p ∞

X

n=1

a^p_n,

where the constant factors_sin(π/p)^π and h π

sin(π/p)

ip

are the best possible. Inequality (1.3) is called Hardy-Hilbert’s inequality, and is important in analysis and its applications (cf. Mitrinovi´c et al. [3]).

In 1997-1998, by estimating the weight coefficient and introducing the Euler constant γ, Yang and Gao [4, 5] gave a strengthened version of (1.3) as:

(1.5)

∞

X

n=1

∞

X

m=1

a_mb_n m+n <







∞

X

n=1



 π sin

π p

− 1−γ n^1p



a^p_n







1 p





∞

X

n=1



 π sin

π p

−1−γ n^1q



b^q_n







1 q

, where1−γ = 0.42278433⁺ is the best value. In 1998, Yang [6] first introduced an indepen- dent parameter λ and the Beta function to build an extension of Hilbert’s integral inequality.

Recently, by introducing a parameterλ,Yang [7] and Yang et al. [8] gave some extensions of (1.3) and (1.4) as: If2−min{p, q}< λ≤2, a_n,b_n≥0satisfy

0<

∞

X

n=1

n^1−λa^p_n<∞ and 0<

∞

X

n=1

n^1−λb^q_n <∞, then one has the following two equivalent inequalities:

(1.6)

∞

X

n=1

∞

X

m=1

a_mb_n

(m+n)^λ < kλ(p) ( _∞

X

n=1

n^1−λa^p_n

)¹_p( _∞ X

n=1

n^1−λb^q_n )¹_q

and (1.7)

∞

X

n=1

n^{(p−1)(λ−1)}

" _∞ X

m=1

a_m (m+n)^λ

#p

dy <[k_λ(p)]^p

∞

X

n=1

n^1−λa^p_n, where the constant factorsk_λ(p) =B

p+λ−2

p ,^q+λ−2_q

and[k_λ(p)]^pare the best possible (B(u, v) is theβfunction). Forλ = 1,inequalities (1.6) and (1.7) reduce respectively to (1.3) and (1.4).

By introducing a parameter α, Kuang [9] gave an extension of (1.3), and Yang [10] gave an improvement of [9] as: If0< α≤min{p, q}, a_n,b_n ≥0satisfy

0<

∞

X

n=1

n^{(p−1)(1−α)}a^p_n<∞ and 0<

∞

X

n=1

n^{(q−1)(1−α)}b^q_n <∞,

then one has two equivalent inequalities as:

(1.8)

∞

X

n=1

∞

X

m=1

a_mb_n

m^α+n^α < π αsin

π p

( _∞

X

n=1

n^{(p−1)(1−α)}a^p_n

)¹_p( _∞ X

n=1

n^{(q−1)(1−α)}b^q_n )¹_q

and (1.9)

∞

X

n=1

n^α−1

" _∞ X

m=1

a_m m^α+n^α

#p

dy <



 π αsin

π p





p ∞

X

n=1

n^{(p−1)(1−α)}a^p_n,

where the constant factors _αsin(π/p)^π and h π

αsin(π/p)

ip

are the best possible. Forα = 1,inequali- ties (1.8) and (1.9) reduce respectively to (1.3) and (1.4). Recently, Hong [11] gave an extension of (1.3) by introducing two parametersλandαas: Ifα≥1,1− _αr¹ < λ≤1 (r=p, q),then (1.10)

∞

X

n=1

∞

X

m=1

a_mb_n

(m^α+n^α)^λ < Hλ,α(p) ( _∞

X

n=1

n^α(1−λ)a^p_n

)¹_p( _∞ X

n=1

n^α(1−λ)b^p_n )¹_q

, where

H_λ,α(p) =

B

1− 1

αq, λ+ 1 αq −1

_p¹ B

1− 1

αp, λ+ 1 αp−1

¹_q .

Forλ=α= 1,(1.10) reduces to (1.3). However, it is obvious that (1.10) is not an extension of (1.6) or (1.8).

In 2003, Yang et al. [12] provided an extensive account of the above results. More recently, Yang [13] gave some extensions of (1.1) and (1.2) as: If0< λ≤min{p, q},satisfy

0<

∞

X

n=1

n^p−1−λa^p_n<∞ and 0<

∞

X

n=1

n^q−1−λa^p_n <∞, then one has the following two equivalent inequalities:

(1.11)

∞

X

n=1

∞

X

m=1

a_mb_n

(m+n)^λ < K_λ(p) ( _∞

X

n=1

n^p−1−λa^p_n

)¹_p( _∞ X

n=1

n^q−1−λb^p_n )¹_q

and (1.12)

∞

X

n=1

n^{(p−1)λ−1}

" _∞ X

m=1

a_m (m+n)^λ

#p

dy <[K_λ(p)]^p

∞

X

n=1

n^p−1−λa^p_n, where the constantsK_λ(p) = B

λ p,^λ_q

and[K_λ(p)]^p are the best possible. For λ = 1,(1.11) and (1.12) reduce to the following two equivalent inequalities:

(1.13)

∞

X

n=1

∞

X

m=1

ambn

m+n < π sin

π p

( _∞

X

n=1

n^p−2a^p_n

)_p¹ ( _∞ X

n=1

n^q−2b^q_n )¹_q

and (1.14)

∞

X

n=1

n^p−2

∞

X

m=1

a_m m+n

!p

<



 π sin

π p





p ∞

X

n=1

n^p−2a^p_n.

Forp = q = 2, inequalities (1.13) and (1.14) reduce respectively to (1.1) and (1.2). We find that inequalities (1.3) and (1.13) are different, although both of them are the best extensions of (1.1) with the(p, q)−parameter.

The main objective of this paper is to obtain some extensions of (1.3) with the best constant factors, by introducing two parameters λ and α and using the Beta function, related to the double series asP∞

n=1

P∞ m=1

ambn

(m^α+n^α)^λ (λ, α > 0),so that inequality (1.10) can be improved.

The equivalent form and some reversions are considered.

2. SOMELEMMAS

First, we need the form of the Beta function as (cf. Wang et al. [14]):

(2.1) B(u, v) :=

Z ∞ 0

1

(1 +t)^u+vt^u−1dt=B(v, u) (u, v >0).

Lemma 2.1. If p > 0 (p 6= 1), ¹_p + ¹_q = 1, λ, α > 0, φr = φr(λ, α) > 0 (r = p, q),satisfy φ_p +φ_q=λα, define the weight functionω_r(x)as

(2.2) ω_r(x) :=

Z ∞ 0

x^λα−φr (x^α+y^α)^λ

1 y

1−φr

dy (x >0; r=p, q).

Then forx >0, eachω_r(x)is constant, that is

(2.3) ω_r(x) = 1

αB φp

α ,φq

α

(x >0;r =p, q).

Proof. Settingu= ^y_xα

in the integral (2.2), one hasdy= ^x_αu^1α−1duand ω_r(x) =x^λα−φr

Z ∞ 0

1 (x^α+x^αu)^λ

1 xu^1/α

1−φr

x

αu^α1−1du

= 1 α

Z ∞ 0

1

(1 +u)^λu^φrα−1du (r=p, q).

By (2.1), sinceφ_p +φ_q=λα, one has (2.3). The lemma is proved.

Lemma 2.2. If p > 1, ¹_p + ¹_q = 1, λ, α > 0, φ_r > 0 (r = p, q),satisfy φ_p +φ_q = λα, and 0< ε < qφ_p,then one has

I₁ :=

Z ∞ 1

Z ∞ 1

x^{−1+φq−εp}

(x^α+y^α)^λy^{−1+φp−εq}dxdy

> 1 εαB

φ_p α − ε

qα,φ_q α + ε

qα

−O(1).

(2.4)

If0< p <1and0< ε <−qφ_q,with the above assumption, then one has I₂ :=

∞

X

m=1

Z ∞ 0

m^{−1+φq−εp}

(m^α+y^α)^λy^{−1+φp−εq}dy

= 1 αB

φ_p α − ε

qα,φ_q α + ε

qα ^∞

X

m=1

1 m^1+ε. (2.5)

Proof. Settingu= ^y_xα

in the integralI₁,one has I₁ =

Z ∞ 1

x^{−1+φq−pε} Z ∞

1

1

(x^α+y^α)^λy^{−1+φp−qε}dy

dx

= 1 α

Z ∞ 1

x^−1−ε Z ∞

1 xα

1

(1 +u)^λu^{φpα−qαε−1}dudx

= 1 εα

Z ∞ 0

u^{φpα−qαε −1}

(1 +u)^λ du− 1 α

Z ∞ 1

x^−1−ε Z _xα¹

0

u^{φpα−qαε−1} (1 +u)^λ dudx

> 1 εα

Z ∞ 0

u^{φpα−qαε −1}

(1 +u)^λ du− 1 α

Z ∞ 1

x⁻¹ Z _xα¹

0

u^{φpα−qαε −1}dudx

= 1 εα

Z ∞ 0

u^{φpα−qαε −1} (1 +u)^λ du−

φp− ε

q −2

. (2.6)

By (2.1), it follows that (2.4) is valid. For0 < p <1,settingu= (_m^y)^α in the integral ofI₂,in

the same manner, one has (2.5). The lemma is thus proved.

3. MAINRESULTS

Theorem 3.1. If p > 1, ¹_p + ¹_q = 1, λ, α > 0, 0 < φr ≤ 1 (r = p, q), φp +φq = λα and a_n, b_n≥0satisfy

0<

∞

X

n=1

n^{p(1−φq)−1}a^p_n <∞ and 0<

∞

X

n=1

n^{q(1−φp)−1}b^q_n <∞, then one has

(3.1)

∞

X

n=1

∞

X

m=1

ambn

(m^α+n^α)^λ < 1 αB

φp

α,φq

α

( _∞ X

n=1

n^{p(1−φq)−1}a^p_n

)¹_p( _∞ X

n=1

n^{q(1−φp)−1}b^q_n )¹_q

, where the constant factor _α¹B

φp

α,^φ_α^q

is the best possible.

Proof. By Hölder’s inequality with weight (see [15]), one has H(am, bn) :=

∞

X

n=1

∞

X

m=1

a_mb_n (m^α+n^α)^λ

=

∞

X

n=1

∞

X

m=1

1 (m^α+n^α)^λ

m^(1−φq)/q

n^(1−φp)/pa_m n^(1−φp)/p m^(1−φq)/qb_n

≤ ( _∞

X

m=1

" _∞ X

n=1

m^λα−φp

(m^α+n^α)^λ · 1 n^1−φp

#

m^{p(1−φq)−1}a^p_m )¹_p

× ( _∞

X

n=1

" _∞ X

m=1

n^λα−φq

(m^α+n^α)^λ · 1 m^1−φq

#

n^{q(1−φp)−1}b^q_n )¹_q

. (3.2)

Sinceλ, α >0,and1−φ_r ≥0 (r =p, q),in view of (2.2), we rewrite (3.2) as H(a_m, b_n)<

( _∞ X

m=1

ω_p(m)m^{p(1−φq)−1}a^p_m

)¹_p( _∞ X

n=1

ω_q(n)n^{q(1−φp)−1}b^q_n )¹_q

,

and then by (2.3), one has (3.1). For 0 < ε < qφ_p, setting a⁰_nand b⁰_n as: a⁰_n = n^{−1+φq−εp}, b⁰_n =n^{−1+φp−εq}, n∈N,then we find

( _∞ X

n=1

n^{p(1−φq)−1}a^0p_n

)¹_p( _∞ X

n=1

n^{q(1−φp)−1}b^0q_n )¹_q

= 1 +

∞

X

n=2

1 n^1+ε (3.3)

<1 + Z ∞

1

1 t^1+εdt

= 1

ε(1 +ε).

If the constant factor _α¹B_φ

p

α,^φ_α^q

in (3.1) is not the best possible, then there exists a positive constantk(withk < ¹_αB

φp

α,^φ_α^q

), such that (3.1) is still valid if one replaces _α¹B φp

α,^φ_α^q

by k. In particular, by (2.4) and (3.3),

1 αB

φ_p α − ε

qα,φ_q α + ε

qα

−εO(1) < εI₁

< εH(a⁰_m, b⁰_n)

< εk ( _∞

X

n=1

n^{p(1−φq)−1}a^0p_n

)¹_p ( _∞ X

n=1

n^{q(1−φp)−1}b^0q_n )¹_q

=k(1 +ε), and then _α¹B_φ

p

α,^φ_α^q

≤ k (ε → 0⁺).This contradicts the fact thatk < _α¹B_φ

p

α,^φ_α^q

.Hence the constant factor ¹_αB

φp

α,^φ_α^q

in (3.1) is the best possible. The theorem is proved.

Theorem 3.2. Ifp > 1, ¹_p+¹_q = 1, λ, α >0,0< φ_r ≤1 (r=p, q),φ_p+φ_q =λαanda_n ≥0 satisfy

0<

∞

X

n=1

n^{p(1−φq)−1}a^p_n<∞, then one has

(3.4)

∞

X

n=1

n^pφp−1

" _∞ X

m=1

am

(m^α+n^α)^λ

#p

<

1 αB

φp

α,φq

α

p ∞

X

n=1

n^{p(1−φq)−1}a^p_n,

where the constant factor h1

αB

φp

α,^φ_α^qip

is the best possible. Inequality (3.4) is equivalent to (3.1).

Proof. Set

b_n:=n^pφp−1

" _∞ X

m=1

a_m (m^α+n^α)^λ

#p−1

,

and use (3.1) to obtain 0<

∞

X

n=1

n^{q(1−φp)−1}b^q_n (3.5)

=

∞

X

n=1

n^pφp−1

" _∞ X

m=1

a_m (m^α+n^α)^λ

#p

=

∞

X

n=1

∞

X

m=1

a_mb_n (m^α+n^α)^λ

≤ 1 αB

φ_p α,φ_q

α

( _∞ X

n=1

n^{p(1−φq)−1}a^p_n

)¹_p( _∞ X

n=1

n^{q(1−φp)−1}b^q_n )¹_q

and

0<

( _∞ X

n=1

n^{q(1−φp)−1}b^q_n )¹_p

= ( _∞

X

n=1

n^pφp−1

" _∞ X

m=1

a_m (m^α+n^α)^λ

#

p

)_p¹

≤ 1 αB

φ_p α,φ_q

α

( _∞ X

n=1

n^{p(1−φq)−1}a^p_n )¹_p

<∞.

(3.6)

It follows that (3.5) takes the form of strict inequality by using (3.1); so does (3.6). Hence, one has (3.4).

On the other hand, if (3.4) is valid, by Hölder’s inequality, one has

∞

X

n=1

∞

X

m=1

a_mb_n (m^α+n^α)^λ =

∞

X

n=1

" _∞ X

m=1

n^1q−1+φpa_m (m^α+n^α)^λ

# h

n^1−φp−1qb_ni (3.7)

≤ ( _∞

X

n=1

n^pφp−1

" _∞ X

m=1

a_m (m^α+n^α)^λ

#p)_p¹ ( _∞ X

n=1

n^{q(1−φp)−1}b^q_n )¹_q

. By (3.4), one has (3.1). It follows that inequalities (3.4) and (3.1) are equivalent. If the constant factor in (3.4) is not the best possible, one can obtain a contradiction that the constant factor in (3.1) is not the best possible by using (3.7). Hence the constant factor in (3.4) is still the best

possible. Thus the theorem is proved.

Theorem 3.3. If0< p < 1, ¹_p +¹_q = 1,

A ={(λ, α); λ, α >0, 0< φ_r≤1 (r=p, q), φ_p+φ_q =λα} 6= Φ, andan, bn ≥0satisfy

0<

∞

X

n=1

n^{p(1−φq)−1}a^p_n <∞ and 0<

∞

X

n=1

n^{q(1−φp)−1}b^q_n <∞,

then for(λ, α)∈A,one has (3.8)

∞

X

n=1

∞

X

m=1

ambn

(m^α+n^α)^λ

> 1 αB

φ_p α,φ_q

α

( _∞ X

n=1

[1−θ_p(n)]n^{p(1−φq)−1}a^p_n

)¹_p( _∞ X

n=1

n^{q(1−φp)−1}b^q_n )¹_q

,

where0< θp(n) =O _n¹_φp

<1;the constant _α¹B φp

α,^φ_α^q

is the best possible.

Proof. By the reverse of Hölder’s inequality (see [15]), following the method of proof in Theo- rem 3.1, since0< p <1andq <0,one has

(3.9) H(a_m, b_n)>

( _∞ X

n=1

$_p(n)n^{p(1−φq)−1}a^p_n

)_p¹ ( _∞ X

n=1

ω_q(n)n^{q(1−φp)−1}b^q_n )¹_q

, whereω_q(n)is defined as in (2.2) and

(3.10) $_p(n) :=

∞

X

k=1

n^λα−φp (n^α+k^α)^λ

1 k

^1−φp

(n ∈N).

Defineθ_p(n)as

(3.11) θ_p(n) := n^λα−φp ω_p(n)

Z 1 0

1 (n^α+y^α)^λ

1 y

1−φp

dy (n ∈N).

Since

ω_p(n)>

Z 1 0

n^λα−φp (n^α+y^α)^λ

1 y

1−φp

dy, then we find0< θp(n)<1,and

(3.12) $_p(n)>

Z ∞ 1

n^λα−φp (n^α+y^α)^λ

1 y

1−φp

dy =ω_p(n) [1−θ_p(n)]. By (3.12), (2.3) and (3.9), one has (3.8). Since

(3.13) 0< θ_p(n)< n^λα−φp ω_p(n)

Z 1 0

1 n^λα

1 y

1−φp

dy= 1

ω_p(n)φ_p · 1 n^φp, andω_p(n)is a constant, we haveθ_p(n) = O _n¹_φp

(n → ∞).

For0 < ε <min{q(φ_p−1),−qφ_q},settinga⁰_nandb⁰_n as: a⁰_n =n^{−1+φq−pε}, b⁰_n =n^{−1+φp−qε}, n∈N,sinceφp >0,then

∞

X

n=1

O 1

n^φp+1+ε

=O(1)(ε→0⁺),

and

( _∞ X

n=1

[1−θ_p(n)]n^{p(1−φq)−1}a^0p_n

)¹_p( _∞ X

n=1

n^{q(1−φp)−1}b^0q_n )¹_q (3.14)

=

∞

X

n=1

1 n^1+ε





 1−

P∞ n=1O

1 n^φp+1+ε

P∞

n=1 1 n^1+ε







1 p

=

∞

X

n=1

1

n^1+ε(1−o(1))^1p. If the constant ¹_αB

φp

α,^φ_α^q

in (3.8) is not the best possible, then there exists a positive number K (withK > _α¹B_φ

p

α,^φ_α^q

), such that (3.8) is still valid if one replaces _α¹B_φ

p

α,^φ_α^q

byK. In particular, by (3.14) and (2.5), one has

K

∞

X

n=1

1

n^1+ε{1−o(1)}^1p

=K ( _∞

X

n=1

[1−θ_p(n)]n^{p(1−φq)−1}a^0p_n

)¹_p( _∞ X

n=1

n^{q(1−φp)−1}b^0q_n )¹_q

< H(a⁰_m, b⁰_n)

< I₂ = 1 αB

φ_p α − ε

qα,φ_q α + ε

qα ^∞

X

n=1

1 n^1+ε, and thenK ≤ _α¹B_φ

p

α,^φ_α^q

(ε→0⁺).By this contradiction we can conclude that the constant

1 αB

φp

α,^φ_α^q

in (3.8) is the best possible. Thus the theorem is proved.

Theorem 3.4. If0< p < 1, ¹_p +¹_q = 1,

A={(λ, α);λ, α >0, 0< φr ≤1 (r=p, q), φp+φq =λα} 6= Φ, anda_n, b_n ≥0satisfy

0<

∞

X

n=1

n^{p(1−φq)−1}a^p_n<∞, for(λ, α)∈A,one has

(3.15)

∞

X

n=1

n^pφp−1

" _∞ X

m=1

a_m (m^α+n^α)^λ

#p

>

1 αB

φ_p α,φ_q

α

p ∞

X

n=1

[1−θ_p(n)]n^{p(1−φq)−1}a^p_n, where0< θ_p(n) =O _n¹_φp

<1,and the constant factorh

1 αB_φ

p

α,^φ_α^qip

is the best possible.

Inequality (3.15) is equivalent to (3.8).

Proof. Still setting

b_n :=n^pφp−1

" _∞ X

m=1

a_m (m^α+n^α)^λ

#p−1

,

by (3.8), one has 0<

∞

X

n=1

n^{q(1−φp)−1}b^q_n (3.16)

=

∞

X

n=1

n^pφp−1

" _∞ X

m=1

a_m (m^α+n^α)^λ

#p

=

∞

X

n=1

∞

X

m=1

a_mb_n (m^α+n^α)^λ

≥ 1 αB

φ_p α,φ_q

α

( _∞ X

n=1

[1−θ_p(n)]n^{p(1−φq)−1}a^p_n

)_p¹ ( _∞ X

n=1

n^{q(1−φp)−1}b^q_n )¹_q

and

0<

( _∞ X

n=1

n^{q(1−φp)−1}b^q_n )¹_p

= ( _∞

X

n=1

n^pφp−1

" _∞ X

m=1

a_m (m^α+n^α)^λ

#p)¹_p

≥ 1 αB

φ_p α,φ_q

α

( _∞ X

n=1

[1−θp(n)]n^{p(1−φq)−1}a^p_n )¹_p

. (3.17)

IfP∞

n=1n^{q(1−φp)−1}b^q_n < ∞, by using (3.8), (3.16) takes the form of strict inequality; so does (3.17). If P∞

n=1n^{q(1−φp)−1}b^q_n = ∞, (3.17) takes naturally strict inequality. Hence we have (3.15).

On the other hand, if (3.15) is valid, by the reverse of Hölder’s inequality,

∞

X

n=1

∞

X

m=1

a_mb_n (m^α+n^α)^λ =

∞

X

n=1

" _∞ X

m=1

n^1q−1+φpa_m (m^α+n^α)^λ

#

[n^1−φp−1qbn] (3.18)

≥ ( _∞

X

n=1

n^pφp−1

" _∞ X

m=1

a_m (m^α+n^α)^λ

#p)¹_p( _∞ X

n=1

n^{q(1−φp)−1}b^q_n )¹_q

. Hence by (3.15), one has (3.8). If the constant factor in (3.15) is not the best possible, we can conclude that the constant factor in (3.8) is not the best possible by using (3.18). The theorem

is proved.

Note: In view of (3.1), ifφ_r =φ_r(λ, α) (r=p, q)satisfy B(φ_p(1,1), φ_q(1,1)) = π

sin

π p

,

andrφ_r(1,1) = 1 (r = p, q),one can get a best extension of (1.3); if B(φ_p(1,1), φ_q(1,1)) =

π

sin(π/p)(or ^π₂),andrφ_r(1,1)6= 1 (r =p, q), one can get a best extension of (1.1) but not a best extension of (1.3). For example, settingφr = ₁

r(α−2) + 1

λ (r = p, q),then rφr(1,1) = r−16= 1,by Theorems 3.1 – 3.4, one can get a best extension of (1.13) and (1.1) as follows:

Corollary 3.5. Ifp >1, ¹_p+¹_q = 1, λ >0, α >2−min{p, q},₁

r(α−2) + 1

λ≤1 (r =p, q), a_n,b_n≥0, satisfy

0<

∞

X

n=1

np[1−λ(α−1)]+(α−2)λ−1

a^p_n<∞ and

0<

∞

X

n=1

nq[1−λ(α−1)]+(α−2)λ−1

b^q_n <∞, one has equivalent inequalities as:

(3.19)

∞

X

n=1

∞

X

m=1

ambn

(m^α+n^α)^λ

< K_λ,α(p)× ( _∞

X

n=1

np[1−λ(α−1)]+(α−2)λ−1

a^p_n

)_p¹ ( _∞ X

n=1

nq[1−λ(α−1)]+(α−2)λ−1

b^q_n )¹_q

and (3.20)

∞

X

n=1

n^{(p+α−2)λ−1}

" _∞ X

m=1

am

(m^α+n^α)^λ

#p

<[K_λ,α(p)]^p

∞

X

n=1

np[1−λ(α−1)]+(α−2)λ−1

a^p_n, where

K_λ,α(p) = 1 αB

λp+α−2

αp , λq+α−2 αq

and[K_λ,α(p)]^p are the best possible. In particular,

(i) forα= 1,one has0< λ≤min{p, q}and (1.11);

(ii) forλ= 1,one has2−min{p, q}< α≤2and (3.21)

∞

X

n=1

∞

X

m=1

a_mb_n

m^α+n^α < K_1,α(p) ( _∞

X

n=1

n(p−1)(2−α)−1

a^p_n

)¹_p( _∞ X

n=1

n(q−1)(2−α)−1

b^q_n )¹_q

and (3.22)

∞

X

n=1

n^p+α−3

" _∞ X

m=1

a_m m^α+n^α

#p

<[K_1,α(p)]^p

∞

X

n=1

n(p−1)(2−α)−1

a^p_n.

If0< p <1,for(λ, α) = (1,2)∈ A(6= Φ),by (3.8), (3.15) and (3.13), one can obtain two equivalent reversions as

(3.23)

∞

X

n=1

∞

X

m=1

a_mb_n

m² +n² > π 2

( _∞ X

n=1

1− 2

πn a^p_n

n

)¹_p( _∞ X

n=1

b^q_n n

)¹_q

and (3.24)

∞

X

n=1

n^p−1

" _∞ X

m=1

a_m m²+n²

#p

>π 2

p ∞

X

n=1

1− 2

πn a^p_n

n , where the constant factors in the above inequalities are the best possible.

4. SOME BESTEXTENSIONS OF(1.3) Settingφr= ^λα_r (r=p, q),by Theorems 3.1 – 3.4, one has

Corollary 4.1. Ifp >1, ¹_p + ¹_q = 1, λ, α > 0, λα≤min{p, q}, a_n, b_n≥0, satisfy 0<

∞

X

n=1

n(p−1)(1−λα)

a^p_n <∞ and 0<

∞

X

n=1

n(q−1)(1−λα)

b^q_n<∞, then one has the following equivalent inequalities:

(4.1)

∞

X

n=1

∞

X

m=1

a_mb_n

(m^α+n^α)^λ < K_λ(p) α

( _∞ X

n=1

n(p−1)(1−λα)

a^p_n

)¹_p( _∞ X

n=1

n(q−1)(1−λα)

b^q_n )¹_q

and (4.2)

∞

X

n=1

n^λα−1

" _∞ X

m=1

a_m (m^α+n^α)^λ

#p

<

K_λ(p) α

p ∞

X

n=1

n(p−1)(1−λα)a^p_n, whereK_λ(p) = B

λ p,^λ_q

.In particular,

(i) forα= 1,one has0< λ≤min{p, q}and the following two equivalent inequalities:

(4.3)

∞

X

n=1

∞

X

m=1

a_mb_n

(m+n)^λ < K_λ(p) ( _∞

X

n=1

n^{(p−1)(1−λ)}a^p_n

)¹_p( _∞ X

n=1

n^{(q−1)(1−λ)}b^q_n )¹_q

and (4.4)

∞

X

n=1

n^λ−1

" _∞ X

m=1

a_m (m+n)^λ

#p

<[K_λ(p)]^p

∞

X

n=1

n^{(p−1)(1−λ)}a^p_n; (ii) forλ= 1,0< α≤min{p, q}one has two equivalent inequalities as:

(4.5)

∞

X

n=1

∞

X

m=1

ambn

m^α+n^α < π αsin

π p

( _∞

X

n=1

n^{(p−1)(1−α)}a^p_n

)¹_p( _∞ X

n=1

n^{(q−1)(1−α)}b^q_n )¹_q

and (4.6)

∞

X

n=1

n^α−1

" _∞ X

m=1

a_m m^α+n^α

#p

<



 π αsin

π p





p ∞

X

n=1

n^{(p−1)(1−α)}a^p_n, where the constant factors in the above inequalities are the best possible.

Note: Since for0 < p < 1, φ_q = ^λα_q < 0,thenA = Φ.It follows that both (4.1) and (4.2) do not possess reversions. Settingφ_r = ^λα−1₂ + ¹_r (r =p, q),by Theorems 3.1 – 3.4, one has Corollary 4.2. Ifp > 1, ¹_p + ¹_q = 1, λ, α > 0,1−2 minn

1 p,¹_qo

< λα ≤ 1 + 2 minn

1 p,¹_qo

, a_n, b_n≥0, satisfy

0<

∞

X

n=1

n^p2(1−λα)a^p_n <∞ and 0<

∞

X

n=1

n^q2(1−λα)b^q_n<∞,

then one has the following equivalent inequalities:

(4.7)

∞

X

n=1

∞

X

m=1

a_mb_n (m^α+n^α)^λ

< 1 αB

pλα−p+ 2

2pα ,qλα−q+ 2 2qα

( _∞ X

n=1

n^p2(1−λα)a^p_n

)¹_p( _∞ X

n=1

n^2q(1−λα)b^q_n )¹_q

and (4.8)

∞

X

n=1

n^p2(λα−1)

" _∞ X

m=1

a_m (m^α+n^α)^λ

#p

dy

<

1 αB

pλα−p+ 2

2pα ,qλα−q+ 2 2qα

p ∞

X

n=1

n^p2(1−λα)a^p_n. In particular,

(i) forα = 1,one has1−2 minn

1 p,¹_qo

< λ≤ 1 + 2 minn

1 p,¹_qo

,and the following two equivalent inequalities:

(4.9)

∞

X

n=1

∞

X

m=1

a_mb_n (m+n)^λ

< B

pλ−p+ 2

2p ,qλ−q+ 2 2q

( _∞ X

n=1

n^p2(1−λ)a^p_n

)¹_p( _∞ X

n=1

n^q2(1−λ)b^q_n )¹_q

and (4.10)

∞

X

n=1

n^p2(λ−1)

" _∞ X

m=1

a_m (m+n)^λ

#p

dy

<

B

pλ−p+ 2

2p ,qλ−q+ 2 2q

p ∞

X

n=1

n^p2(1−λα)a^p_n; (ii) for λ = 1, one has 1− 2 minn

1 p,¹_qo

< α ≤ 1 + 2 minn

1 p,¹_qo

and two equivalent inequalities as:

(4.11)

∞

X

n=1

∞

X

m=1

a_mb_n m^α+n^α

< 1 αB

pα−p+ 2

2pα ,qα−q+ 2 2qα

( _∞ X

n=1

n^p2(1−α)a^p_n

)¹_p( _∞ X

n=1

n^q2(1−α)b^q_n )¹_q

and (4.12)

∞

X

n=1

n^p2(α−1)

" _∞ X

m=1

am

m^α+n^α

#p

dy

<

1 αB

pα−p+ 2

2pα ,qα−q+ 2 2qα

p ∞

X

n=1

n^p2(1−α)a^p_n, where the constant factors in the above inequalities are the best possible.