Some new Hardy-Hilbert-type inequalities with multiparameters

The purpose of this paper is to build some new Hardy-Hilbert-type inequalities with multiparameters and their equivalent forms and variants, which generalize some existing results. Similarly, the corresponding Hardy-Hilbert-type integral inequalities are also given.


Introduction
Let a n , b n ≥ 0, p > 1, 1/p + 1/q = 1. If 0 < ∞ n=1 a p n < ∞ and 0 < ∞ n=1 b q n < ∞, then where the constant π/ sin(π/p) is the best possible. The inequality (1.1) can be called as the well known Hardy-Hilbert's inequality [1]. An equivalent form of inequality (1.1) is presented as follows. where the constant [π/ sin(π/p)] p is also the best possible. In connection with applications in analysis, their generalizations and variants have received considerable interest recent years [2][3][4][5][6][7][8][9][10][11][12]. By introducing some parameters, Yang [13] obtained a generalization of Hardy-Hilbert's integral inequality with a best constant factor that involves the beta function. In the paper [14], Das and sahoo considered a generalization of multiple Hardy-Hilbert's inequality with the best constant factor. Sroysang [15] established a generalization on the kinds of Hardy-Hilbert's integral inequality with the weight homogeneous function.
In the paper, motivated by the mentioned references above, we will obtain a Hardy-Hilbert-type inequality with multiparameters, which can be see as a new generalization of (1.3)-(1.5). And its equivalent form and variant are given. Furthermore, their integral forms are also presented.
The idea of proof of Theorem 2.1 comes similarly from [3] and [9]. To prove the Theorem 2.1, we need some lemmas in the same way, which are new generalizations of some lemmas given in [8,9].
Similar to Lemma 2.1, we can introduce the following lemma.
Nextly, we will give the proof of Theorem 2.1.
Proof. By using the Hölder's inequality and Lemmas 2.1 and 2.2, we can capture which implies (2.1). Now we prove that the constant c λ,p is the best possible. Suppose that the constant c λ,p in inequality (2.1) is not the best possible, then there exists positive k < c λ,p such that k replaces with c λ,p in inequality (2.1) holds all the same. Especially, for ε ∈ (0, q/2p), let a n = (1/n) On the other hand, we have It follows from (2.4) and (2.5) that we observe as ε tends to 0 + . According to the assumption, we have where in the third equality we set that u = y λ 2 /x λ 1 . At the same time, when ε tends to 0 + , we have which implies that if ε tends to 0 + , the following equations holds true It follows from Lemma 2.3 that when ε tends to 0 + , we get Due to the equations (2.6) and (2.7), we can obtain as ε tends to 0 + . We have c λ,p < k, in contradiction with supposition. So the constant c λ,p in inequality (2.1) is the best possible. This finishes the proof of Theorem 2.1.
Theorem 2.2. Let a m ≥ 0,and p > 1, where the constant c p λ,p is the best possible. Moreover, inequality (2.8) is equivalent to inequality (2.1).
On the other hand, if inequality (2.8) holds, by applying the Hölder's inequality, we have which means that inequality (2.1) holds. Since the constant c λ,p in inequality (2.1) is the best possible, and inequality (2.1) is the equivalent to inequality (2.8), the constant c λ,p in inequality (2.8) is also the best possible. This completes the proof of Theorem 2.2.
Now we present a variant of Theorem 2.1 as follows.
When a m , b n are replaced by a m /m, b n /n in inequality (2.9), respectively, it is easy to obtain that inequality (2.9) is equivalent to the following new variant of Muholland's inequality.

From the above inequalities, we can obtain
According to (2.10) and (2.11), then we give which impies c λ,p < k, in contradiction with supposition. So the constant c λ,p in inequality (2.9) is the best possible. This completes the proof of Theorem 2.3.

New corresponding Hardy-Hilbert-type integral inequalities
In this section, we will give new double integral forms of double series inequalities.
where the constant factors in the inequalities (3.1) and (3.2) are also the best possible. Moreover, the inequalities (3.1) and (3.2) are equivalent.
Proof. By mean of Hölder's inequality and Lemmas 2.1 and 2.2, we have According to the hypotheses, it is easy to see that the equality in the above the second inequality is not possible. Now we prove that the constant c λ,p is the best possible. If the constant c λ,p is not the best possible, then there exists k < c λ,p such that k replaces with c λ,p , inequality (3.1) holds all the same. Especially, for ε ∈ (0, q/(2p)), we set Based on the course of the proof of Theorem 2.1, we have as ε tends to 0 + . We have c λ,p < k, in contradiction with supposition. So the constant c λ,p in the inequality (3.1) is the best possible. There exists t 0 > 0 such that , which implies that as T tends to ∞, then we have which means the inequality (3.2) holds. If inequality (3.2) holds, by using the Hölder's inequality, then which implies that the inequality (3.1) holds. Therefore, inequalities (3.1) and (3.2) are equivalent. It is easy to see that the constant c p λ,p in the inequality (3.2) is also the best possible. Now we present a new variant of Theorem 3.1 as follows.