A New Hilbert-Type Inequality with Positive Homogeneous Kernel and Its Equivalent Forms

: We establish a new inequality of Hilbert-type containing positive homogeneous kernel ( min { m , n } ) λ and derive its equivalent forms. Based on the obtained Hilbert-type inequality, we discuss its equivalent forms and give the operator expressions in some particular cases.

The following analogue of Hardy-Hilbert's inequality is known in the literature as Hardy-Littlewood-Polya's inequality, and the constant factor pq in (3) is the best possible (c.f. [1], Theorem 341).
Recently, by applying inequality (3), Adiyasuren, Batbold and Azar [15] gave a new Hilber-type inequality with the kernel 1 (m+n) λ and partial sums. In 2016, Hong and Wen [16] studied the equivalent statements of the extended inequalities (1) and (2), and estimated the best possible constant factor for several parameters.
Motivated by the ideas of [2] and [16], in the present paper we deal with a new Hilbert-type inequality containing positive homogeneous kernel (min{m, n}) λ and deduce its equivalent forms. Furthermore, we discuss the equivalent statements relating to the best possible constant factor, based on the obtained Hilbert-type inequality.
(ii) For λ 1 ∈ (1, 11 8 ] ∩ (0, λ), by using the Euler-Maclaurin summation formula (c.f. [2,3]) with the Bernoulli function of 1-order ρ(t) and then one has Thus, we get On the other hand, we have and then by 1 2 − Hence, from the expression g m (t) we deduce the inequalities in (7). The proof of Lemma 1 is thus complete.
Next, we shall establish a new inequality of Hilbert type for positive homogeneous kernel.

Remark 1. By inequality
and the following inequality: In Lemma 3 below, we show that the constant factor given in (10) is the best possible.
Proof. For any 0 < ε < qλ 1 , we set If there exists a constant M ≤ λ λ 1 λ 2 such that (10) is valid when replacing λ λ 1 λ 2 by M, then in particular, by substitution of a m = a m and b n = b n in (10), we have In the following, we shall prove that M ≥ λ λ 1 λ 2 , which would reveal that M = λ λ 1 λ 2 is the best possible constant factor in (10).

Main Results and Some Particular Cases
Theorem 1. Inequality (8) is equivalent to the following inequality: If the constant factor in (8) is the best possible, then so is the constant factor in (14). (14) is valid. By Hölder's inequality (c.f. [25]), we have

Proof. Suppose that inequality
Then, by using inequality (14), we obtain inequality (8). On the other hand, assuming that inequality (8) is valid, we set (14) is naturally valid; if J = ∞, then it is impossible to make inequality (14) valid, which implies J < ∞. Suppose that 0 < J < ∞. By inequality (8), we have ∞ n=1 n q(1+ Thus, inequality (14) follows, and we conclude that inequality (8) is equivalent to inequality (14). Furthermore, we show that if the constant factor in (8) is the best possible, then the constant factor in (14) is also the best possible. Otherwise, from inequality (15) we would reach a contradiction, namely that the constant factor in (8) is not the best possible. The proof of Theorem 1 is thus completed.
Therefore, we assert that the statements (i), (ii), (iii) and (iv) are equivalent. This completes the proof of Theorem 2. Now, we discuss some particular cases of the inequalities obtained above, from which we will derive some interesting inequalities. (10) and (16), we obtain the following equivalent inequalities with the best possible constant factor pq: (10) and (16), we get the following equivalent inequalities with the best possible constant factor pq: (iii) Setting p = q = 2, both (17) and (19) reduce to the inequality: furthermore, both (18) and (20) reduce to the equivalent form of (21) as follows: (iv) Putting λ = 2, λ 1 = λ 2 = 1 in (10) and (16), we have the following equivalent inequalities with the best possible constant factor 2: (v) Putting λ = e (< 11 4 = 2.75), λ 1 = λ 2 = e 2 in (10) and (16), we have the following equivalent inequalities with the best possible constant factor 4 e :

Operator Expressions
We choose the functions where from, We define the following real normed spaces: We let a ∈ l p,ϕ , and set (min{m, n}) λ a m , n ∈ N.

Conclusions
In this paper, we give, with Lemma 2 and Theorem 1, respectively, a new inequality of the Hilbert-type containing positive homogeneous kernel and its equivalent forms. Based on the obtained Hilbert-type inequality, we discuss in Theorem 2 the equivalent statements of the best possible constant factor related to several parameters. As applications, the operator expressions of the obtained inequalities are given in Theorem 3, and some particular cases of the obtained inequalities (10) and (16) are considered in Remark 2. It is shown that the results obtained in Theorems 1 and 2 would generate more new inequalities of Hilbert-type.
Author Contributions: B.Y. carried out the mathematical studies and drafted the manuscript. S.W. and A.W. participated in the design of the study and performed the numerical analysis. All authors contributed equally and significantly in this paper. All authors have read and agreed to the published version of the manuscript.