Open Access

On reverse Hilbert-type inequalities

Journal of Inequalities and Applications20142014:198

https://doi.org/10.1186/1029-242X-2014-198

Received: 8 November 2013

Accepted: 6 May 2014

Published: 20 May 2014

Abstract

By introducing two pairs of conjugate exponents and estimating the weight coefficients, we establish reverse versions of Hilbert-type inequalities, as described by Jin (J. Math. Anal. Appl. 340:932-942, 2008), and we prove that the constant factors are the best possible. As applications, some particular results are considered.

Keywords

reverse Hilbert-type inequality weight coefficient best constant factor

1 Introduction

If both a n and b n 0 , such that 0 < n = 1 a n 2 < and 0 < n = 1 b n 2 < , then we have (see [1])
n = 1 m = 1 a m b n m + n < π { n = 1 a n 2 } 1 2 { n = 1 b n 2 } 1 2 ,
(1.1)

where the constant factor π has the best possible value. Inequality (1.1) is the well-known Hilbert inequality, introduced in 1925; inequality (1.1) has been generalized by Hardy as follows.

If p > 1 , 1 p + 1 q = 1 , and both a n and b n 0 , such that 0 < n = 1 a n p < and 0 < n = 1 b n q < , then we have
n = 1 m = 1 a m b n m + n < π sin ( π / p ) { n = 1 a n p } 1 p { n = 1 b n q } 1 q ,
(1.2)

where the constant factor π sin ( π / p ) is the best possible. Inequality (1.2) is the well-known Hardy-Hilbert inequality, which is important in analysis and applications (see [2]). In recent years, many results with generalizations of this type of inequality have been obtained (see [3]).

Under the same conditions as in (1.2), there are some Hilbert-type inequalities that are similar to (1.2), which also have been studied and generalized by some mathematicians.

Recently, by studying a Hilbert-type operator, Jin [4] obtained a new bilinear operator inequality with the norm, and he provided some new Hilbert-type inequalities with the best constant factor. First, we repeat the results of [4].

Definition 1.1 Let H p , q ( r , s ) be the set of functions k ( x , y ) satisfying the following conditions.

Let p > 1 , 1 p + 1 q = 1 , r > 1 , 1 r + 1 s = 1 , suppose that k ( x , y ) is continuous in ( 0 , ) × ( 0 , ) and satisfies:
  1. (1)

    k ( x , y ) = k ( y , x ) > 0 , where x , y ( 0 , ) .

     
  2. (2)

    For ε 0 and x > 0 , the function k ( x , t ) ( x t ) 1 + ε l ( l = r , s ) is decreasing in t ( 0 , ) .

     
For ε 0 small enough, x > 0 , and k ¯ l ( ε , x ) can be written as
k ¯ l ( ε , x ) : = 0 k ( x , t ) ( x t ) 1 + ε l d t ( l = r , s ) ,
where k ¯ l ( ε , x ) is independent of x, k ¯ l ( 0 , x ) : = 0 k ( x , t ) ( x t ) 1 l d t = k p ( l = r , s ), k p is a positive constant independent of x, and k ¯ l ( ε , x ) = k p ( ε ) = k p + o ( 1 ) ( ε 0 + ).
( 3 ) m = 1 1 m 1 + ε 0 1 k ( m , t ) ( m t ) 1 + ε ( s / q ) s d t = O ( 1 ) ( ε 0 + ) , ( 3 ) m = 1 1 m 1 + ε 0 1 k ( m , t ) ( m t ) 1 + ε ( r / p ) r d t = O ( 1 ) ( ε 0 + ) .

We have Jin’s result as follows.

Theorem 1.1 If p > 1 , 1 p + 1 q = 1 , r > 1 , 1 r + 1 s = 1 , and k ( x , y ) H p , q ( r , s ) , a n , b n 0 , such that 0 < n = 1 n p r 1 a n p < and 0 < n = 1 n q s 1 b n q < , then we have
n = 1 m = 1 k ( m , n ) a m b n < k r { n = 1 n p r 1 a n p } 1 p { n = 1 n q s 1 b n q } 1 q ,
(1.3)
n = 1 n p r 1 [ m = 1 k ( m , n ) a m ] p < ( k r ) p n = 1 n p r 1 a n p .
(1.4)

Here the constant factors k r and ( k r ) p are the best possible. Inequality (1.3) is equivalent to (1.4).

If p = r and q = s in Theorem 1.1, then Theorem 1.1 reduces to Yang’s result [5] as follows.

Theorem 1.2 If p > 1 , 1 p + 1 q = 1 , k ( x , y ) H ( p , q ) , and both a n and b n 0 , such that 0 < n = 1 a n p < and 0 < n = 1 a n p < , then we have
n = 1 m = 1 k ( m , n ) a n b n < k p { n = 1 a n p } { n = 1 b n q } 1 q ,
(1.5)
n = 1 [ n = 1 k ( m , n ) a m ] p < ( k p ) p n = 1 a n p ,
(1.6)

where the constant factors k p and ( k p ) p are the best possible. Inequality (1.3) is equivalent to (1.4).

In this paper, by introducing some parameters, we establish a reverse version of the inequality (1.3). As applications, some particular results are considered.

2 Some lemmas

Definition 2.1 Let H p , q ( r , s ) be the set of functions k ( x , y ) satisfying the following conditions:

Let 0 < p < 1 , 1 p + 1 q = 1 , r > 1 , 1 r + 1 s = 1 , suppose that k ( x , y ) is continuous in ( 0 , ) × ( 0 , ) and satisfies:
  1. (1)

    k ( x , y ) = k ( y , x ) > 0 , x , y ( 0 , ) .

     
  2. (2)

    For ε 0 and x > 0 , the function k ( x , t ) ( x t ) 1 + ε l ( l = r , s ) is decreasing in t ( 0 , ) .

     
For ε 0 small enough, for x > 0 , k ¯ l ( ε , x ) can be described as
k ¯ l ( ε , x ) : = 0 k ( x , t ) ( x t ) 1 + ε l d t ( l = r , s ) ,
where k ¯ l ( ε , x ) is independent of x, and k ¯ l ( 0 , x ) : = 0 k ( x , t ) ( x t ) 1 l d t = k p ( l = r , s ), k p is a positive constant independent of x, and k ¯ l ( ε , x ) = k p ( ε ) = k p + o ( 1 ) ( ε 0 + ).
  1. (3)
    There exists a positive constant λ such that
    θ λ ( s , m ) = 1 k r 0 1 k ( m , t ) ( m t ) 1 s d t = O ( 1 / m λ ) ( 0 , 1 ) ( m ) , θ λ ( r , n ) = 1 k r 0 1 k ( t , n ) ( n t ) 1 r d t = O ( 1 / n λ ) ( 0 , 1 ) ( n ) .
     
Lemma 2.2 If 0 < p < 1 , 1 p + 1 q = 1 , r > 1 , 1 r + 1 s = 1 , k ( x , y ) H p , q ( r , s ) , and the weight coefficients w ( r , p , m ) and w ( s , q , n ) are defined as
ω ( r , p , m ) = n = 1 k ( m , n ) m p 1 r n 1 s ,
(2.1)
ω ( s , q , n ) = m = 1 k ( m , n ) n q 1 s m 1 r ,
(2.2)
then we have
m p r 1 k r ( 1 θ λ ( s , m ) ) < ω ( r , p , m ) < m p r 1 k r , n q s 1 k r ( 1 θ λ ( r , n ) ) < ω ( s , q , n ) < n q s 1 k r .
(2.3)
Proof By the assumption of the lemma, because k ( x , t ) ( x t ) 1 s ( t ( 0 , ) ) is decreasing, then we find
ω ( r , p , m ) = m p r 1 n = 1 k ( m , n ) ( m n ) 1 s m p r 1 0 k ( m , t ) ( m t ) 1 s d t = m p r 1 k r .
However, we find
ω ( r , p , m ) = m p r 1 n = 1 k ( m , n ) ( m n ) 1 s m p r 1 1 k ( m , t ) ( m t ) 1 s d t = m p r 1 0 k ( m , t ) ( m t ) 1 s d t m p r 1 0 1 k ( m , t ) ( m t ) 1 s d t = m p r 1 k r m p r 1 0 1 k ( m , t ) ( m t ) 1 s d t = m p r 1 k r ( 1 1 k r 0 1 k ( m , t ) ( m t ) 1 s d t ) = m p r 1 k r ( 1 θ λ ( s , m ) ) .

It is easy to show that the above inequalities take the form of a strict inequality. Hence, we have m p r 1 k r ( 1 θ λ ( s , m ) ) < ω ( r , p , m ) < m p r 1 k r . Similarly, we can obtain n q s 1 k r ( 1 θ λ ( r , n ) ) < ω ( s , q , n ) < n q s 1 k r . The lemma is proved. □

Lemma 2.3 If 0 < p < 1 , 1 p + 1 q = 1 , r > 1 , 1 r + 1 s = 1 , and k ( x , y ) H p , q ( r , s ) , for ε > 0 small enough, we have
m = 1 n = 1 k ( m , n ) m 1 r ε p n 1 s ε q < ( k r + o ( 1 ) ) 1 m 1 ε ( ε 0 + ) .
(2.4)
Proof For ε > 0 , by Definition 2.1, we have
m = 1 n = 1 k ( m , n ) m 1 r ε p n 1 s ε q = m = 1 m 1 ε ε n = 1 k ( m , n ) ( m n ) 1 + ε ( s / q ) s m = 1 m 1 ε 0 k ( m , t ) ( m t ) 1 + ε ( s / q ) s d t = m = 1 m 1 ε k r ( ε s q ) = ( k r + o ( 1 ) ) 1 m 1 ε .

The lemma is proved. □

3 Main results

Theorem 3.1 If 0 < p < 1 , 1 p + 1 q = 1 , r > 1 , 1 r + 1 s = 1 , and k ( x , y ) H p , q ( r , s ) , a n , b n 0 , such that 0 < n = 1 n p r 1 a n p < and 0 < n = 1 n q s 1 b n q < , then we have
n = 1 m = 1 k ( m , n ) a m b n > k r { n = 1 [ 1 θ λ ( s , n ) ] n p r 1 a n p } 1 p { n = 1 n q s 1 b n q } 1 q ,
(3.1)
n = 1 n p r 1 [ m = 1 k ( m , n ) a m ] p > ( k r ) p n = 1 [ 1 θ λ ( s , n ) ] n p r 1 a n p ,
(3.2)

where the constant factor k r and ( k r ) p are the best possible. Inequality (3.1) is equivalent to (3.2).

Proof By Hölder’s inequality, we have (see [6])
n = 1 m = 1 k ( m , n ) a m b n = n = 1 m = 1 { [ k ( m , n ) ] 1 q m 1 q r n 1 p s a m } { [ k ( m , n ) ] 1 q n 1 p s m 1 q r b n } { m = 1 n = 1 k ( m , n ) m p 1 r n 1 s a m p } 1 p { n = 1 m = 1 k ( m , n ) n q 1 s m 1 r b n q } 1 q = { m = 1 ω ( r , p , m ) a m p } 1 p { n = 1 ω ( s , q , n ) b n q } 1 q .
(3.3)

Then, by (2.3), in view of 0 < p < 1 and q < 0 , we have (3.1).

For ε > 0 , setting a ¯ n = n 1 r ε q and b ¯ n = n 1 s ε q , we find
{ n = 1 [ 1 θ λ ( s , n ) ] n p r 1 a ¯ n p } 1 p { n = 1 n q s 1 b ¯ n q } 1 q = { n = 1 n 1 ε n = 1 O ( 1 / n λ ) n 1 ε } 1 p { n = 1 n 1 ε } 1 q = n = 1 n 1 ε [ 1 ( n = 1 n 1 ε ) 1 n = 1 O ( 1 / n λ ) n 1 ε ] 1 p .
(3.4)
By virtue of (2.4), we have
n = 1 m = 1 k ( m , n ) a ¯ m b ¯ n = m = 1 n = 1 k ( m , n ) m 1 r ε p n 1 s ε q < ( k r + o ( 1 ) ) 1 m 1 ε ( ε 0 + ) .
(3.5)
If the constant factor k r in (3.1) is not the best possible factor, then there exists a positive number K (with K > k r ), such that (3.1) is still valid if the constant factor k r is replaced by K. In particular, by (3.4) and (3.5), we have
( k r + o ( 1 ) ) 1 n 1 ε > K n = 1 n 1 ε [ 1 ( n = 1 n 1 ε ) 1 n = 1 O ( 1 / n λ ) n 1 ε ] 1 p ,
that is,
( k r + o ( 1 ) ) > K [ 1 ( n = 1 n 1 ε ) 1 n = 1 O ( 1 / n λ ) n 1 ε ] 1 p .

For ε 0 + , it follows that K k r , which contradicts the fact that K > k r . Hence, the constant factor k r in (3.1) is the best possible.

Setting b n as
b n : = n p r 1 [ m = 1 k ( m , n ) a m ] p 1 ,
by (3.1), we have
{ n = 1 n q s 1 b n q } p = { n = 1 n p r 1 [ m = 1 k ( m , n ) a m ] p } p = { n = 1 m = 1 k ( m , n ) a m b n } p ( k r ) p { n = 1 [ 1 θ λ ( s , n ) ] n p r 1 a n p } { n = 1 n q s 1 b n q } p 1 .
(3.6)
Hence, we obtain
> n = 1 n p r 1 [ m = 1 k ( m , n ) a m ] p ( k r ) p n = 1 [ 1 θ λ ( s , n ) ] n p r 1 a n p > 0 .
(3.7)

By (3.1), both (3.6) and (3.7) take the form of a strict inequality, and we have (3.2).

However, if (3.2) is valid, by Hölder’s inequality, we find
n = 1 m = 1 k ( m , n ) a m b n = n = 1 [ n 1 q 1 s m = 1 k ( m , n ) a m ] [ n 1 s 1 q b n ] { n = 1 n p r 1 [ m = 1 k ( m , n ) a m ] p } 1 p { n = 1 n q s 1 b n q } 1 q .
(3.8)

Then, by (3.2), we have (3.1). Hence (3.2) and (3.1) are equivalent.

If the constant factor ( k r ) p in (3.2) is not the best possible, by using (3.8), we find the contradiction that the constant factor k r in (3.1) is not the best possible. The theorem is completed. □

4 Some particular results

  1. (1)
    Setting
    k ( x , y ) = ( x y ) λ 1 2 ( x + y ) λ ( 1 2 min { 1 r , 1 s } < λ 1 + 2 min { 1 r , 1 s } ) ,
     
for 0 ε < min { p ( λ + 1 2 1 r ) , q ( λ + 1 2 1 s ) } , and for fixed x > 0 , we find (see [4])
k ¯ s ( ε , x ) B ( s ( λ + 1 ) 2 2 s , r ( λ + 1 ) 2 2 r ) = k r ( ε 0 + ) ,
and k ¯ s ( ε , x ) k r ( ε 0 + );
0 < 0 1 k ( m , t ) ( m t ) 1 s d t = 0 1 ( m t ) λ 1 2 ( m + t ) λ ( m t ) 1 s d t 0 1 ( m t ) λ 1 2 m λ ( m t ) 1 s d t = 1 ( λ 1 2 + 1 r ) 1 m 1 + λ 2 1 s .
Hence, θ λ ( s , m ) = O ( m 1 s 1 + λ 2 ) . Similarly, we obtain θ λ ( r , n ) = O ( n 1 r 1 + λ 2 ) . For ε 0 , 1 2 min { 1 r , 1 s } < λ 1 + 2 min { 1 r , 1 s } , and fixed x > 0 , the function
k ( x , t ) ( x t ) 1 + ε l = ( x t ) λ 1 2 ( x + t ) λ ( x t ) 1 + ε l = x 1 + ε l + λ 1 2 ( x + t ) λ t λ 1 2 1 + ε l ( l = r , s )

is decreasing in ( 0 , ) . Hence, k ( x , y ) H p ( r , s ) . By Theorem 3.1, we have the following.

Corollary 4.1 If 0 < p < 1 , 1 / p + 1 / q = 1 , 1 / r + 1 / s = 1 , 1 2 min { 1 r , 1 s } < λ 1 + 2 min { 1 r , 1 s } , and both a n , b n 0 such that 0 < n = 1 n p r 1 a n p < and 0 < n = 1 n q s 1 b n q < , then we have
n = 1 m = 1 ( m n ) λ 1 2 ( m + n ) λ a m b n > B ( s ( λ + 1 ) 2 2 s , r ( λ + 1 ) 2 2 r ) { n = 1 [ 1 θ λ ( s , n ) ] n p r 1 a n p } 1 p { n = 1 n q r 1 b n q } 1 q ,
(4.1)
n = 1 n p r 1 [ m = 1 ( m n ) λ 1 2 ( m + n ) λ a m ] p > [ B ( s ( λ + 1 ) 2 2 s , r ( λ + 1 ) 2 2 r ) ] p n = 1 [ 1 θ λ ( s , n ) ] n p r 1 a n p ,
(4.2)
where the constant factors
B ( s ( λ + 1 ) 2 2 s , r ( λ + 1 ) 2 2 r ) and [ B ( s ( λ + 1 ) 2 2 s , r ( λ + 1 ) 2 2 r ) ] p

are the best possible. Inequality (4.1) is equivalent to (4.2).

In particular, (a) for r = q , s = p , and 1 2 min { 1 p , 1 q } < λ 1 + 2 min { 1 p , 1 q } , we have
n = 1 m = 1 ( m n ) λ 1 2 ( m + n ) λ a m b n > B ( p ( λ + 1 ) 2 2 p , q ( λ + 1 ) 2 2 q ) { n = 1 [ 1 θ λ ( p , n ) ] n p 2 a n p } 1 p { n = 1 n q 2 b n q } 1 q ,
(4.3)
n = 1 n p 2 [ m = 1 ( m n ) λ 1 2 ( m + n ) λ a m ] p > [ B ( p ( λ + 1 ) 2 2 p , q ( λ + 1 ) 2 2 q ) ] p n = 1 [ 1 θ λ ( p , n ) ] n p 2 a n p .
(4.4)
  1. (b)
    For r = s = 2 and 0 < λ 2 , we have
    n = 1 m = 1 ( m n ) λ 1 2 ( m + n ) λ a m b n > B ( λ 2 , λ 2 ) { n = 1 [ 1 θ λ ( 2 , n ) ] n p 2 1 a n p } 1 p { n = 1 n q 2 1 b n q } 1 q ,
    (4.5)
    n = 1 n p 2 1 [ m = 1 ( m n ) λ 1 2 ( m + n ) λ a m ] p > [ B ( λ 2 , λ 2 ) ] p n = 1 [ 1 θ λ ( 2 , n ) ] n p 2 1 a n p .
    (4.6)
     
  2. (2)
    Let
    k ( x , y ) = ( x y ) λ 1 2 x λ + y λ ( 1 2 min { 1 r , 1 s } < λ 1 + 2 min { 1 r , 1 s } ) .
     
For 0 ε < min { p ( λ + 1 2 1 r ) , q ( λ + 1 2 1 s ) } and x > 0 , we find (see [4])
k ¯ s ( ε , x ) 1 λ B ( s ( λ + 1 ) 2 2 s λ , r ( λ + 1 ) 2 2 r λ ) = k r ( ε 0 + ) ,
and k ¯ s ( ε , x ) k r ( ε 0 + );
0 1 k ( m , t ) ( m t ) 1 s d t = 0 1 ( m t ) λ 1 2 m λ + t λ ( m t ) 1 s d t 0 1 ( m t ) λ 1 2 m λ ( m t ) 1 s d t = 1 ( λ 1 2 + 1 r ) m 1 s 1 + λ 2 .
Hence, θ λ ( s , m ) = O ( m 1 s 1 + λ 2 ) . Similarly, we can obtain θ λ ( r , n ) = O ( n 1 r 1 + λ 2 ) . For ε 0 , 1 2 min { 1 r , 1 s } < λ 1 + 2 min { 1 r , 1 s } , and x > 0 , the function
k ( x , t ) ( x t ) 1 + ε l = ( x t ) λ 1 2 x λ + t λ ( x t ) 1 + ε l = x 1 + ε l + λ 1 2 x λ + t λ t λ 1 2 1 + ε l ( l = r , s )

is decreasing in ( 0 , ) . Hence k ( x , y ) H p ( r , s ) . By Theorem 3.1, we have the following corollary.

Corollary 4.2 If 0 < p < 1 , 1 p + 1 q = 1 , r > 1 , 1 r + 1 s = 1 , 1 2 min { 1 r , 1 s } < λ 1 + 2 min { 1 r , 1 s } , and both a n , b n 0 such that 0 < n = 1 n p r 1 a n p < and 0 < n = 1 n q s 1 b n q < , then we have
n = 1 m = 1 ( m n ) λ 1 2 m λ + n λ a m b n > 1 λ B ( s ( λ + 1 ) 2 2 s λ , r ( λ + 1 ) 2 2 r λ ) { n = 1 [ 1 θ λ ( s , n ) ] n p r 1 a n p } 1 p { n = 1 n q r 1 b n q } 1 q ,
(4.7)
n = 1 n p r 1 [ m = 1 ( m n ) λ 1 2 m λ + n λ a m ] p > [ 1 λ B ( s ( λ + 1 ) 2 2 s λ , r ( λ + 1 ) 2 2 r λ ) ] p n = 1 [ 1 θ λ ( s , n ) ] n p r 1 a n p ,
(4.8)
where the constant factors
1 λ B ( s ( λ + 1 ) 2 2 s λ , r ( λ + 1 ) 2 2 r λ ) and [ 1 λ B ( s ( λ + 1 ) 2 2 s λ , r ( λ + 1 ) 2 2 r λ ) ] p

are the best possible. Inequality (4.7) is equivalent to (4.8).

In particular, (a) for r = q , s = p , and 1 2 min { 1 p , 1 q } < λ 1 + 2 min { 1 p , 1 q } , we have
n = 1 m = 1 ( m n ) λ 1 2 m λ + n λ a m b n > 1 λ B ( p ( λ + 1 ) 2 2 p λ , q ( λ + 1 ) 2 2 q λ ) × { n = 1 [ 1 θ λ ( p , n ) ] n p 2 a n p } 1 p { n = 1 n q 2 b n q } 1 q ,
(4.9)
n = 1 n p 2 [ m = 1 ( m n ) λ 1 2 ( m + n ) λ a m ] p > [ 1 λ B ( p ( λ + 1 ) 2 2 p λ , q ( λ + 1 ) 2 2 q λ ) ] p n = 1 [ 1 θ λ ( p , n ) ] n p 2 a n p .
(4.10)
  1. (b)
    For r = s = 2 and 0 < λ 2 , we have
    n = 1 m = 1 ( m n ) λ 1 2 m λ + n λ a m b n > π λ { n = 1 [ 1 θ λ ( 2 , n ) ] n p 2 1 a n p } 1 p { n = 1 n q 2 1 b n q } 1 q ,
    (4.11)
    n = 1 n p 2 1 [ m = 1 ( m n ) λ 1 2 m λ + n λ a m ] p > [ π λ ] p n = 1 [ 1 θ λ ( 2 , n ) ] n p 2 1 a n p .
    (4.12)
     
  2. (3)
    Let
    k ( x , y ) = ( x y ) λ 1 2 ( max { x , y } ) λ ( 1 2 min { 1 r , 1 s } < λ 1 + 2 min { 1 r , 1 s } ) ,
     
for 0 ε < min { p ( λ + 1 2 1 r ) , q ( λ + 1 2 1 s ) } and x > 0 , then we find (see [4])
k ¯ s ( ε , x ) 4 r s λ [ r ( λ + 1 ) 2 ] [ s ( λ + 1 ) 2 ] = k r ( ε 0 + ) ,
and k ¯ r ( ε , x ) k r ( ε 0 + )
0 < 0 1 k ( m , t ) ( m t ) 1 s d t = 0 1 ( m t ) λ 1 2 ( max { m , t } ) λ ( m t ) 1 s d t = 0 1 ( m t ) λ 1 2 m λ ( m t ) 1 s d t = 1 ( λ 1 2 + 1 r ) 1 m 1 + λ 2 1 s .
Hence, θ λ ( s , m ) = O ( 1 m 1 + λ 2 1 s ) . Similarly, we can obtain θ λ ( r , n ) = O ( 1 n 1 + λ 2 1 r ) . For ε 0 , 1 2 min { 1 r , 1 s } < λ 1 + 2 min { 1 r , 1 s } , and x > 0 , the function
k ( x , t ) ( x t ) 1 + ε l = ( m t ) λ 1 2 ( max { m , t } ) λ ( x t ) 1 + ε l = x 1 + ε l + λ 1 2 ( max { m , t } ) λ t λ 1 2 1 + ε l ( l = r , s )

is decreasing in ( 0 , ) . Hence, k ( x , y ) H p , q ( r , s ) . By Theorem 3.1, we have the following corollary.

Corollary 4.3 If 0 < p < 1 , 1 p + 1 q = 1 , r > 1 , 1 r + 1 s = 1 , 1 2 min { 1 r , 1 s } < λ 1 + 2 min { 1 r , 1 s } , and both a n , b n 0 , such that 0 < n = 1 n p r 1 a n p < and 0 < n = 1 n q s 1 b n q < , then we have
n = 1 m = 1 ( m n ) λ 1 2 ( max { m , n } ) λ a m b n > 4 r s λ [ r ( λ + 1 ) 2 ] [ s ( λ + 1 ) 2 ] { n = 1 [ 1 θ λ ( s , n ) ] n p r 1 a n p } 1 p { n = 1 n q s 1 b n q } 1 q ,
(4.13)
n = 1 n p r 1 [ ( m n ) λ 1 2 ( max { m , n } ) λ a m ] p > ( 4 r s λ [ r ( λ + 1 ) 2 ] [ s ( λ + 1 ) 2 ] ) p n = 1 [ 1 θ λ ( s , n ) ] n p r 1 a n p .
(4.14)

Here the constant factors 4 r s λ [ r ( λ + 1 ) 2 ] [ s ( λ + 1 ) 2 ] and ( 4 r s λ [ r ( λ + 1 ) 2 ] [ s ( λ + 1 ) 2 ] ) p are the best possible. Inequality (4.13) is equivalent to (4.14).

In particular, (a) for r = q , s = p , and 1 2 min { 1 p , 1 q } < λ 1 + 2 min { 1 p , 1 q } , we have
n = 1 m = 1 ( m n ) λ 1 2 ( max { m , n } ) λ a m b n > 4 p q λ [ p ( λ + 1 ) 2 ] [ q ( λ + 1 ) 2 ] { n = 1 [ 1 θ λ ( p , n ) ] n p 2 a n p } 1 p { n = 1 n q 2 b n q } 1 q ,
(4.15)
n = 1 n p 2 [ m = 1 ( m n ) λ 1 2 ( max { m , n } ) λ a m ] p > ( 4 p q λ [ p ( λ + 1 ) 2 ] [ q ( λ + 1 ) 2 ] ) p n = 1 [ 1 θ λ ( p , n ) ] n p 2 a n p .
(4.16)
  1. (b)
    For r = s = 2 and 0 < λ 2 , we have
    n = 1 m = 1 ( m n ) λ 1 2 ( max { m , n } ) λ a m b n > 4 λ { n = 1 [ 1 θ λ ( 2 , n ) ] n p 2 1 a n p } 1 p { n = 1 n q 2 1 b n q } 1 q ,
    (4.17)
    n = 1 n p 2 1 [ m = 1 ( m n ) λ 1 2 ( max { m , n } ) λ a m ] p > ( 4 λ ) p n = 1 [ 1 θ λ ( 2 , n ) ] n p 2 1 a n p .
    (4.18)
     

Declarations

Acknowledgements

The work is supported by Scientific Research Program Funded by Shaanxi Provincial Education Department (No. 2013JK1139), China Postdoctoral Science Foundation (No. 2013M542370), NNSFC (No. 11326161), key projects of Science and Technology Research of the Henan Education Department (No. 14A110011). The authors deeply appreciate the support.

Authors’ Affiliations

(1)
School of Mathematical Science, Huaibei Normal University
(2)
Department of Education Science, Pingxiang University
(3)
School of Computer Science and Engineering, Xi’an University of Technology
(4)
Department of Electrical and Computer Engineering, Cornell University

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© Xu et al.; licensee Springer. 2014

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