ON A REVERSE HARDY-LITTLEWOOD-PÓLYA’S INEQUALITY∗

Abstract By the use of the weight coefficients, the idea of introduced parameters and Euler-Maclaurin summation formula, a reverse Hardy-LittlewoodPólya’s inequality with parameters as well as the equivalent forms are provided. The equivalent statements of the best possible constant factor related to a few parameters and some particular cases are given.

In 1934, a half-discrete Hilbert-type inequality was given as follows (cf. [4], In the last ten years, some extensions of (1.5) with their applications and the reverses were provided by [16-18, 25, 26]. In 2016, by means of the technique of real analysis, Hong et al. [7] considered some equivalent statements of the extensions of (1.1) with the best possible constant factor related to a few parameters. The other similar works about (1.2), (1.4) and (1.5) were given by [8-11, 22, 27].
In this paper, following the methods of [12] and [7], by the use of the weight coefficients, the idea of introducing parameters and Euler-Maclaurin summation formula, a reverse Hardy-Littlewood-Pólya's inequality with parameters as well as the equivalent forms are provided in Lemma 2.2 and Theorem 3.1. The equivalent statements of the best possible constant factor related to a few parameters and some particular cases are considered in Theorem 3.2 and Remark 3.1.
and then it follows that and then we obtain On the other hand, we find and then for 1 Hence, in view of the result of the case (i), for λ 2 ∈ (1, 11 8 ] ∩ (0, λ), we still have (2.2).
The lemma is proved.

Lemma 2.2.
We have the following reverse Hardy-littlewood-Polya's inequality with parameters: Proof. In the same way of obtaining (2.2), for n ∈ N, we have the following inequality of the weight coefficient: By the reverse Hölder's inequality (cf. [14]), we obtain Then by (2.2) and (2.4), for 0 < p < 1, q < 0, we have (2.3). The lemma is proved.
If there exists a constant M ≥ λ λ1λ2 , such that (2.5) is valid when replacing λ λ1λ2 by M , then in particular, substitution of a m = a m and b n = b n in (2.5), we have By (2.5) and the decreasingness property of series, we obtain By (2.4), setting Then we have For ε → 0 + , we find λ λ1λ2 ≥ M . Hence, M = λ λ1λ2 is the best possible constant factor in (2.5).

Lemma 2.4. If inequality (2.7) is valid with the best possible constant factor
By the reverse Hölder's inequality, we obtain (2.8) , namely, (2.8) keeps the form of equality. We observe that (2.8) keeps the form of equality if and only if there exist constants A and B, such that they are not all zero and (cf. [14]) Assuming that A ̸ = 0, we have u λ−λ1−λ2 = B A a.e. in R + , and then λ − λ 1 − λ 2 = 0, namely, λ = λ 1 + λ 2 .
The lemma is proved.

If the constant factor in (2.3) is the best possible, then so is the constant factor in (3.1) and (3.2).
Proof. Suppose that (3.1) is valid. By the reverse Hölder's inequality, we have Then by (3.1), we obtain (2.3). On the other hand, assuming that (2.3) is valid, we set a m (max{m, n}) λ ] p−1 , n ∈ N.

Conclusions
In this paper, by the use of the weight coefficients, the idea of introduced parameters and Euler-Maclaurin summation formula, a reverse Hardy-Littlewood-Pólya's inequality with parameters and the equivalent forms are given in Lemma 2.2 and Theorem 3.1. The equivalent statements of the best possible constant factor related to a few parameters, and some particular cases are considered in Theorem 3.2 and Remark 3.1. The lemmas and theorems provide an extensive account of this type of inequalities.