A New Reverse Extended Hardy–Hilbert’s Inequality with Two Partial Sums and Parameters

: By using the methods of real analysis and the mid-value theorem, we introduce some lemmas and obtain a new reverse extended Hardy–Hilbert’s inequality with two partial sums and multi-parameters. We also give a few equivalent conditions of the best possible constant factor related to several parameters in the new inequality. Some particular inequalities are deduced.

In this article, based on the idea of [17,18], by using the techniques of real analysis and the mid-value theorem, we introduce some preserving lemmas and then give a reverse of (2) with two partial sums and multi-parameters, which is a new reverse version of the inequality in [16].We also consider a few equivalent conditions of the best possible constant factor in the reverse inequality related to multi-parameters.Furthermore, several inequalities are deduced by setting some particular parameters.

Some Lemmas
In what follows, we assume that 0 < p < 1 (q < 0), 1  p , and the following inequalities: Lemma 1.For t > 0, the following inequalities on the partial sums are valid: Proof.Since A m e −tm α = o(1)(m → ∞) , by Abel's summation by parts formula, it follows that A m e −tm α − e −t(m+1) α .
The lemma is proved.
The lemma is proved.
The lemma is proved.

Main Results
By Lemma 1 and Lemma 4, the following theorem follows: Theorem 1.The following reverse inequality with two partial sums and parameters is valid: In particular, for λ 1 + λ 2 = λ, we have as well as: Proof.Based on the expression as follows 1 6) and ( 7), we have Then, by (15), inequality (17) follows.
The theorem is proved.
In the following two theorems, we provide a few equivalent conditions on (17).17) is the best possible.constant factor.
Setting ε → 0 + in the above inequality, in virtue of the continuity of the beta function, we find is the best possible constant factor in (18).
The theorem is proved.

Conclusions
In this article, by using the techniques of real analysis, the way of weight coefficients and the idea of introduced parameters, applying the mid-value theorem.We estimate some lemmas and obtain a new reverse extended inequality (2) with two partial sums and multi-parameters in Theorem 1.We consider a few equivalent statements of the best possible constant factor related to several parameters in Theorems 2 and 3. We also deduce