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volume 6, issue 3, article 81, 2005.

Received 23 February, 2005;

accepted 17 June, 2005.

Communicated by:W.S. Cheung

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Journal of Inequalities in Pure and Applied Mathematics

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

BICHENG YANG

Department of Mathematics Guangdong Institute of Education Guangzhou, Guangdong 510303 People’s Republic of China

EMail:bcyang@pub.guangzhou.gd.cn

c

2000Victoria University ISSN (electronic): 1443-5756 055-05

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On Best Extensions of Hardy-Hilbert’s Inequality with

Two Parameters Bicheng Yang

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J. Ineq. Pure and Appl. Math. 6(3) Art. 81, 2005

Abstract

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

2000 Mathematics Subject Classification:26D15.

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

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 (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

ambn

m+n < π sin

π p

( _∞

X

n=1

a^p_n

)¹_p( _∞ X

n=1

b^q_n )¹_q

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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 andh

π sin(π/p)

ip

are the best possible. Inequal- ity (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 constantγ, 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 n¹^p



a^p_n







1 p





∞

X

n=1



 π sin

π p

−1−γ n¹^q



b^q_n







1 q

,

where 1 − γ = 0.42278433⁺ is the best value. In 1998, Yang [6] first in- troduced an independent parameter λ and the Beta function to build an extension of Hilbert’s integral inequality. Recently, by introducing a parameter λ, Yang [7] and Yang et al. [8] gave some extensions of (1.3) and (1.4) as: If 2−min{p, q}< λ≤2, an,bn≥0satisfy

0<

∞

Xn^1−λa^p_n <∞ and 0<

∞

Xn^1−λb^q_n<∞,

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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)]^p are the best possible (B(u, v) is the β function). For λ = 1, inequalities (1.6) and (1.7) reduce respectively to (1.3) and (1.4). By introducing a parameterα,Kuang [9]

gave an extension of (1.3), and Yang [10] gave an improvement of [9] as: If 0< 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

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and (1.9)

∞

X

n=1

n^α−1

" _∞ X

m=1

am

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 _α_sin(π/p)^π and h π

αsin(π/p)

ip

are the best possible. For α = 1, inequalities (1.8) and (1.9) reduce respectively to (1.3) and (1.4). Re- cently, Hong [11] gave an extension of (1.3) by introducing two parametersλ andαas: 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

< H_λ,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

¹_p 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: If 0 <

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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

)¹_p( _∞ X

n=1

n^q−2b^q_n )¹_q

and (1.14)

∞

X

n=1

n^p−2

∞

X

m=1

am

m+n

!p

<



 π sin

π p





p ∞

X

n=1

n^p−2a^p_n.

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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 parametersλandαand using the Beta function, related to the double series as P∞

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.

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2. Some Lemmas

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, φ_r = φ_r(λ, α) > 0 (r = p, q),satisfyφp +φq=λα, define the weight functionωr(x)as

(2.2) ωr(x) :=

Z ∞ 0

x^λα−φ^r (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 φ_p

α,φ_q α

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

Proof. Settingu= ^y_xα

in the integral (2.2), one hasdy= _α^xu^α¹⁻¹duand ω_r(x) =x^λα−φ^r

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^φr^α⁻¹du (r =p, q).

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

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Lemma 2.2. If p > 1, ¹_p + ¹_q = 1, λ, α > 0, φ_r > 0 (r = p, q), satisfy φp+φq =λα, and0< ε < qφp,then one has

I₁ :=

Z ∞ 1

x^−1+φ^q⁻^p^ε

(x^α+y^α)^λy^−1+φ^p⁻^ε^qdxdy

> 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+φ^q⁻^ε^p

(m^α+y^α)^λy^−1+φ^p⁻^ε^qdy

= 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+φ^q⁻^ε^p Z ∞

1

(x^α+y^α)^λy^−1+φ^p⁻^ε^qdy

dx

= 1 α

Z ∞ 1

x^−1−ε Z ∞

1 xα

1 (1 +u)^λu

φp α−_qα^ε −1

dudx

= 1 εα

Z ∞ 0

u^φp^α⁻^qα^ε ⁻¹

(1 +u)^λ du− 1 α

Z ∞ 1

x^−1−ε Z _xα¹

0

u^φp^α⁻^qα^ε ⁻¹ (1 +u)^λ dudx

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> 1 εα

Z ∞ 0

u

φp α−_qα^ε −1

(1 +u)^λ du− 1 α

Z ∞ 1

x⁻¹ Z _xα¹

0

u^φp^α⁻^qα^ε⁻¹dudx

= 1 εα

Z ∞ 0

u^φp^α⁻^qα^ε ⁻¹ (1 +u)^λ du−

φ_p −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.

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3. Main Results

Theorem 3.1. If p > 1, ¹_p + ¹_q = 1, λ, α > 0, 0 < φ_r ≤ 1 (r = p, q), φp+φq =λαandan, bn≥0satisfy

0<

∞

X

n=1

n^p(1−φ^q⁾⁻¹a^p_n<∞ and 0<

∞

X

n=1

n^q(1−φ^p⁾⁻¹b^q_n <∞, then one has

(3.1)

∞

X

n=1

∞

X

m=1

< 1 αB

φp

α,φq

α

( _∞ X

n=1

n^p(1−φ^q⁾⁻¹a^p_n

)¹_p( _∞ X

n=1

n^q(1−φ^p⁾⁻¹b^q_n )¹_q

,

where the constant factor _α¹B_φ

p

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

=

∞

X

n=1

∞

X

m=1

1 (m^α+n^α)^λ

m^(1−φ^q^)/q

n^(1−φ^p^)/pa_m n^(1−φ^p^)/p m^(1−φ^q^)/qb_n

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≤ ( _∞

X

m=1

" _∞ X

n=1

m^λα−φ^p

(m^α+n^α)^λ · 1 n^1−φ^p

#

m^p(1−φ^q⁾⁻¹a^p_m )¹_p

× ( _∞

X

n=1

" _∞ X

m=1

n^λα−φ^q

(m^α+n^α)^λ · 1 m^1−φ^q

#

n^q(1−φ^p⁾⁻¹b^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−φ^q⁾⁻¹a^p_m

)¹_p( _∞ X

n=1

ω_q(n)n^q(1−φ^p⁾⁻¹b^q_n )¹_q

, and then by (2.3), one has (3.1). For 0 < ε < qφ_p,settinga⁰_nandb⁰_nas: a⁰_n = n^−1+φ^q⁻^p^ε, b⁰_n=n^−1+φ^p⁻^ε^q, n∈N,then we find

( _∞ X

n=1

n^p(1−φ^q⁾⁻¹a^0p_n

)¹_p( _∞ X

n=1

n^q(1−φ^p⁾⁻¹b^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 < _α¹B

φp

α,^φ_α^q

), such that (3.1) is still valid

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if one replaces _α¹B φp

α,^φ_α^q

byk. 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−φ^q⁾⁻¹a^0p_n

)_p¹ ( _∞ X

n=1

n^q(1−φ^p⁾⁻¹b^0q_n )¹_q

=k(1 +ε), and then _α¹B_φ

p

α,^φ_α^q

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

1 αB

φp

α,^φ_α^q

.Hence the constant factor _α¹B

φp

α,^φ_α^q

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

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

0<

∞

X

n=1

n^p(1−φ^q⁾⁻¹a^p_n<∞, then one has

(3.4)

∞

Xn^pφ^p⁻¹

" _∞

X a_m (m^α+n^α)^λ

#p

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<

1 αB

φ_p α,φ_q

α

p ∞

X

n=1

n^p(1−φ^q⁾⁻¹a^p_n,

where the constant factorh

1 αB

φp

α,^φ_α^qip

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

Proof. Set

b_n:=n^pφ^p⁻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−φ^p⁾⁻¹b^q_n (3.5)

=

∞

X

n=1

n^pφ^p⁻¹

" _∞ X

m=1

a_m (m^α+n^α)^λ

#p

=

∞

X

n=1

∞

X

m=1

≤ 1 αB

φ_p α,φ_q

α

( _∞ X

n=1

n^p(1−φ^q⁾⁻¹a^p_n

)¹_p( _∞ X

n=1

n^q(1−φ^p⁾⁻¹b^q_n )¹_q

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and

0<

( _∞ X

n=1

n^q(1−φ^p⁾⁻¹b^q_n )¹_p

= ( _∞

X

n=1

n^pφ^p⁻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

n^p(1−φ^q⁾⁻¹a^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 (3.7)

=

∞

X

n=1

" _∞ X

m=1

n¹^q^−1+φ^pa_m (m^α+n^α)^λ

# h

n^1−φ^p⁻¹^qb_ni

≤ ( _∞

X

n=1

n^pφ^p⁻¹

" _∞ X

m=1

a_m (m^α+n^α)^λ

#p)¹_p( _∞ X

n=1

n^q(1−φ^p⁾⁻¹b^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).

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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< φr ≤1 (r=p, q), φp+φq =λα} 6= Φ, andan, bn≥0satisfy

0<

∞

X

n=1

n^p(1−φ^q⁾⁻¹a^p_n<∞ and 0<

∞

X

n=1

n^q(1−φ^p⁾⁻¹b^q_n <∞, then for(λ, α)∈A,one has

(3.8)

∞

X

n=1

∞

X

m=1

> 1 αB

φ_p α,φ_q

α

( _∞ X

n=1

[1−θ_p(n)]n^p(1−φ^q⁾⁻¹a^p_n

)¹_p( _∞ X

n=1

n^q(1−φ^p⁾⁻¹b^q_n )¹_q

,

where0< θ_p(n) = O _n¹_φp

<1;the constant_α¹B_φ

p

α,^φ_α^q

is the best possible.

Proof. By the reverse of Hölder’s inequality (see [15]), following the method of proof in Theorem3.1, since0< p <1andq <0,one has

(3.9) H(am, bn)>

( _∞ X

n=1

$p(n)n^p(1−φ^q⁾⁻¹a^p_n

)_p¹ ( _∞ X

n=1

ωq(n)n^q(1−φ^p⁾⁻¹b^q_n )¹_q

,

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whereω_q(n)is defined as in (2.2) and (3.10) $_p(n) :=

∞

X

k=1

n^λα−φ^p (n^α+k^α)^λ

1 k

1−φp

(n ∈N).

Defineθ_p(n)as

(3.11) θ_p(n) := n^λα−φ^p 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−φ^p (n^α+y^α)^λ

1 y

^1−φp

dy, then we find0< θ_p(n)<1,and

(3.12) $_p(n)>

Z ∞ 1

n^λα−φ^p (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< θp(n)< n^λα−φ^p ω_p(n)

Z 1 0

1 n^λα

1 y

1−φp

dy = 1

ω_p(n)φ_p · 1 n^φ^p, 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+φ^q⁻^ε^p, b⁰_n=n^−1+φ^p⁻^ε^q, n∈N,sinceφ_p >0,then

∞

XO 1

n^φ^p+1+ε

=O(1) (ε→0⁺),

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and

( _∞ X

n=1

[1−θ_p(n)]n^p(1−φ^q⁾⁻¹a^0p_n

)¹_p( _∞ X

n=1

n^q(1−φ^p⁾⁻¹b^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))¹^p. If the constant _α¹B_φ

p

α,^φ_α^q

in (3.8) is not the best possible, then there exists a positive numberK (withK > _α¹B

φp

α,^φ_α^q

), such that (3.8) is still valid if one replaces _α¹B

φp

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)}¹^p

=K ( _∞

X

n=1

[1−θ_p(n)]n^p(1−φ^q⁾⁻¹a^0p_n

)¹_p( _∞ X

n=1

n^q(1−φ^p⁾⁻¹b^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,

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and thenK ≤ _α¹B φp

α,^φ_α^q

(ε →0⁺).By this contradiction we can conclude that the constant _α¹B

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< φ_r ≤1 (r =p, q), φ_p+φ_q =λα} 6= Φ, andan, bn≥0satisfy

0<

∞

X

n=1

n^p(1−φ^q⁾⁻¹a^p_n<∞, for(λ, α)∈A,one has

(3.15)

∞

X

n=1

n^pφ^p⁻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−φ^q⁾⁻¹a^p_n, where 0 < θ_p(n) = O _n¹_φp

< 1, and the constant factor h

1 αB_φ

p

α,^φ_α^qip

is the best possible. Inequality (3.15) is equivalent to (3.8).

Proof. Still setting

bn:=n^pφ^p⁻¹

" _∞

X a_m (m^α+n^α)^λ

#^p−1 ,

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by (3.8), one has 0<

∞

X

n=1

n^q(1−φ^p⁾⁻¹b^q_n (3.16)

=

∞

X

n=1

n^pφ^p⁻¹

" _∞ X

m=1

a_m (m^α+n^α)^λ

#p

=

∞

X

n=1

∞

X

m=1

≥ 1 αB

φ_p α,φ_q

α

( _∞ X

n=1

[1−θ_p(n)]n^p(1−φ^q⁾⁻¹a^p_n )¹_p

× ( _∞

X

n=1

n^q(1−φ^p⁾⁻¹b^q_n )¹_q

and

0<

( _∞ X

n=1

n^q(1−φ^p⁾⁻¹b^q_n )¹_p

= ( _∞

X

n=1

n^pφ^p⁻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−θ_p(n)]n^p(1−φ^q⁾⁻¹a^p_n )¹_p

. (3.17)

If P∞

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

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

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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 (3.18)

=

∞

X

n=1

" _∞ X

m=1

n¹^q^−1+φ^pa_m (m^α+n^α)^λ

#

[n^1−φ^p⁻¹^qb_n]

≥ ( _∞

X

n=1

n^pφ^p⁻¹

" _∞ X

m=1

a_m (m^α+n^α)^λ

#p)¹_p( _∞ X

n=1

n^q(1−φ^p⁾⁻¹b^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

,

and rφ_r(1,1) = 1 (r = p, q), one can get a best extension of (1.3); if B(φp(1,1), φq(1,1)) = _sin(π/p)^π (or ^π₂),andrφr(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, setting φ_r = ₁

r(α−2) + 1

λ (r = p, q),then rφ_r(1,1) = r−1 6= 1,by Theorems 3.1–3.4, one can get a best extension of (1.13) and (1.1) as follows:

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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), an,bn ≥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

a_mb_n

(m^α+n^α)^λ < K_λ,α(p)

× ( _∞

X

n=1

np[1−λ(α−1)]+(α−2)λ−1

a^p_n )¹_p

× ( _∞

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

a_m (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,