On an Extension of a Hardy–Hilbert-Type Inequality with Multi-Parameters

Making use of weight coefficients as well as real/complex analytic methods, an extension of a Hardy–Hilbert-type inequality with a best possible constant factor and multiparameters is established. Equivalent forms, reverses, operator expression with the norm, and a few particular cases are also considered.

Then, by (6), we derive that (cf. [14]) where the constant factor is the best possible. Some other kinds of results, such as Hilbert-type integral inequalities, half-discrete Hilbert-type inequalities, and multidimensional Hilbert-type inequalities are provided in .
In the present paper, making use of weight coefficients as well as real/complex analytic methods, a Hardy-Hilbert-type inequality with a best possible constant factor and multiparameters is established (for p > 1). This inequality constitutes an extension of (4) and (7). Equivalent forms, reverses (two cases of 0 < p < 1 and p < 0), operator expression with the norm, and a few particular cases are also considered.

Some Lemmas
In this section we prove the inequalities of the weight functions, which are used to prove the main results. In the sequel, we assume for the multiparameters that , U m and V n are defined by (3), a m , b n ≥ 0 (m, n ∈ N),

Lemma 1.
If C is the set of complex numbers and C ∞ = C ∪ {∞}, are different points, the function f (z) is analytic in C ∞ except for z i (i = 1, 2, · · · , n), and z = ∞ is a zero point of f (z) whose order is not less than 1, then for α ∈ R, we have where 0 < Im(ln z) = arg z < 2π.
In particular, if z k (k = 1, · · · , n) are all poles of order 1, setting Proof. By [43] (p. 118), we obtain (9). We have that In particular, since it is obvious that Then, by (9), we obtain (10). This completes the proof of the lemma.
Since V(y) is strictly increasing in (n − 1, n], λ s > 0 and 1 − λ 2 ≥ 0, in view of the decreasing property, we obtain that Hence, we deduce (15) and (16). This completes the proof of the lemma. where We obtain and then Hence, we deduce (17). Similarly, we obtain (18). For a > 0, we have that Hence, we derive (19). Similarly, we also get (20). This completes the proof of the lemma.
The constant factor k s (λ 1 ) in (22) is still the best possible. Otherwise, we would reach a contradiction by (25) that the constant factor in (21) is not the best possible.
This completes the proof of the theorem.
we define the following normed spaces: we can rewrite (22) as: namely, c ∈ l p,Ψ 1−p λ . Definition 1. Define a Hilbert-type operator T : l p,Φ λ → l p,Ψ 1−p λ as follows: For any a = {a m } ∞ m=1 ∈ l p,Φ λ , there exists a unique representation Ta = c ∈ l p,Ψ 1−p λ . Define the formal inner product of Ta and b = {b n } ∞ n=1 ∈ l q,Ψ λ as follows: We can express the above results in operator forms as: Define the norm of the operator T as follows: Then, by (31), we get that ||T|| ≤ k s (λ 1 ). Since the constant factor in (31) is the best possible, we have ||T|| = k s (λ 1 ).
Suppose that 0 < J < ∞. By (32), it follows that and then (33) follows, which is equivalent to (32). For ε ∈ (0, pλ 1 ), we set λ 1 , λ 2 , a m and b n as in (28). Then by (19), (20) and (16), we find If there exists a positive constant K ≥ k s (λ 1 ), such that (32) is valid when we replace k s (λ 1 ) by K, then in particular, we have namely, It follows that k s (λ 1 ) ≥ K (ε → 0 + ). Hence, K = k s (λ 1 ) is the best possible constant factor of (32). The constant factor k s (λ 1 ) in (33) is still the best possible. Otherwise, we would reach a contradiction by the reverse of (25) that the constant factor in (32) is not the best possible.
This completes the proof of the theorem.
The constant factor k s (λ 1 ) in (37) is still the best possible. Otherwise, we would reach a contradiction by (39) that the constant factor in (36) is not the best possible.
This completes the proof of the theorem.

Conclusions
In the present paper, making use of weight coefficients as well as real/complex analytic methods, a Hardy-Hilbert-type inequality with a best possible constant factor and multiparameters and the equivalent forms are established in Theorems 1 and 2. Reverses, operator expression with the norm, and a few particular cases are also considered in Theorems 3 and 4, Definition 1, and Remark 1. The lemmas and theorems provide an extensive account of this type of inequality.