A REVERSE MORE ACCURATE HARDY-HILBERT’S INEQUALITY ∗

By means of the weight coefficients, the idea of introduced parameters, Hermite-Hadamard’s inequality and Euler-Maclaurin summation formula, a reverse more accurate Hardy-Hilbert’s inequality and the equivalent forms are given. The equivalent statements of the best possible constant factor related to a few parameters are also considered, and some particular reverse inequalities are obtained.

In 2016, by means of the techniques of real analysis, Hong et al. [9] considered some equivalent statements of the extensions of (1) with the best possible constant factor related to several parameters.The other similar works were given by [4,[10][11][12]23].
In this paper, following the way of [9,13], by the use of the weight coefficients, the idea of introduced parameters, Hermite-Hadamard's inequality, Euler-Maclaurin summation formula and the techniques of real analysis, a reverse more accurate Hardy-Hilbert's inequality as well as the equivalent forms are given.The equivalent statements of the best possible constant factor related to several parameters are considered, and and some particular reverse inequalities are obtained.

Some lemmas
In what follows, we assume that 0 < p < 1(q < 0), We also assume that a m , b n ≥ 0, such that Lemma 2.1.For λ 2 ∈ [0, 3  2 ] ∩ (0, λ), define the following weight coefficient: We have the following inequalities: where, O( Proof.For fixed m ∈ N, we set the following real function: In the following we divide two cases to prove (2.3).
On the other hand, we also have where, H(m) is indicated as We have obtained For λ ∈ (0, 3], λ 2 ∈ (1, 3  2 ] ∩ (0, λ), by Euler-Maclaurin summation formula (cf.[13]), we obtain Hence, we have and the following inequalities: In view of the the results in the case (i), we still can obtain (2.3).
The lemma is proved.
The lemma is proved.
Proof.For any 0 < ε < pλ 1 , we set If there exists a constant M ≥ B(λ 1 , λ 2 ), such that (2.6) is valid when we replace B(λ 1 , λ 2 ) by M , then in particular, substitution of a m = a m and b n = b n in (2.6), we have By the decreasingness property of series, we obtain By (2.5), setting Then we have For ε → 0 + , in view of the continuity of the beta function, we find B(λ 1 , λ 2 ) ≥ M .Hence, M = B(λ 1 , λ 2 ) is the best possible constant factor of (2.6).
The lemma is proved.
4), we find )), we obtain and then we have Hence, in view of (i), (ii) and (iii), we still can reduce (2.6) to the following: ) ) is the best possible, then in view of (2.8) and (2.7), we have the following inequality: By the reverse Hölder's inequality (cf.[15]), we find (2.9) Hence, we have B , namely, (2.9) keeps the form of equality.
We observe that (2.9) keeps the form of equality if and only if there exist constants A and B, such that they are not both zero and (cf.[15]) Assuming that A ̸ = 0, it follows that u λ−λ2−λ1 = B/A a.e. in R + ., and then The lemma is proved.

Main results and some particular inequalities
Theorem 3.1.Inequality (2.4) is equivalent to the following inequalities: If the constant factor in (2.4) is the best possible, then, so is the constant factor in (3.1)

Conclusions
In this paper, by means of the weight coefficients, the idea of introduced parameters, Hermite-Hadamard's inequality and Euler-Maclaurin summation formula, a reverse more accurate Hardy-Hilbert's inequality as well as the equivalent forms are given in Lemma 2.2 and Theorem 3.1.The equivalent statements of the best possible constant factor related to several parameters are considered in Theorem 3.2, and some particular reverse inequalities are obtained in Remark 3.1.The lemmas and theorems provide an extensive account of this type of inequalities.