An extension of a multidimensional Hilbert-type inequality

In this paper, by the use of weight coefficients, the transfer formula and the technique of real analysis, a new multidimensional Hilbert-type inequality with multi-parameters and a best possible constant factor is given, which is an extension of some published results. Moreover, the equivalent forms, the operator expressions and a few particular inequalities are considered.


Introduction
 p > , b q > , then we have the following Hardy-Hilbert inequality with the best possible constant π sin(π /p) : ∞ m= ∞ n= a m b n m + n < π sin(π/p) a p b q , (  ) and the following Hilbert-type inequality: (  ) For μ i = ν j =  (i, j ∈ N), inequality () reduces to (). In , Yang and Chen [] gave the following multidimensional Hilbert-type inequality: For i  , j  ∈ N, α, β > , where m = ∞ m i  = · · · ∞ m  = , n = ∞ n j  = · · · ∞ n  = , the series on the right-hand side are positive, and the best possible constant factor K .
In , Shi and Yang [] gave another extension of () as follows: Some other results on Hardy-Hilbert-type inequalities were given by [-]. In this paper, by the use of weight coefficients, the transfer formula and the technique of real analysis, a new multidimensional Hilbert-type inequality with multi-parameters and a best possible constant factor is given, which is an extension of () and (). Moreover, the equivalent forms, the operator expressions and a few particular inequalities are considered.
Then by () and the above result, we find Hence, we have (). In the same way, we have ().
we define two weight coefficients w(λ  , n) and W (λ  , m) as follows: Example  With regard to the assumptions of Definition , we set Then, (i) for fixed y > , is decreasing in R + and strictly decreasing in ([y] + , ∞). In the same way, for fixed x > , Lemma  With regard to the assumptions of Definition , (i) we have Proof (i) By (), () and Example (ii), for  < λ  + η ≤ i  , λ > , it follows that Hence, we have (). In the same way, we have ().

By (), it follows that
Hence, we have and then () and () follow.

Theorem  With regard to the assumptions of Theorem
Then by () and (), we obtain By () and (), we find For ε →  + , we find is the best possible constant factor of (). The constant factor in () is still the best possible. Otherwise, we would reach a contradiction by () that the constant factor in () is not the best possible.

Operator expressions
With regard to the assumptions of Theorem , in view of we can set the following definition.
Definition  Define a multidimensional Hilbert's operator T : l p, → l p, -p as follows: For any a ∈ l p, , there exists a unique representation Ta = c ∈ l p, -p , satisfying Remark  (i) For μ i = ν j =  (i, j ∈ N), () reduces to (). Hence, () is an extension of ().