On a reverse extended Hardy–Hilbert’s inequality

By the use of the weight coefficients, the idea of introducing parameters and the Euler–Maclaurin summation formula, a reverse extended Hardy–Hilbert inequality and the equivalent forms are given. The equivalent statements of the best possible constant factor related to a few parameters and some particular cases are also considered.

In 2016, by means of the techniques of real analysis, Hong et al. [21] considered some equivalent statements of the extensions of (1) with the best possible constant factor related to a few parameters. Similar work about Hilbert-type integral inequalities is in [22][23][24][25][26].
In this paper, following the way of [2,21], by the use of the weight coefficients, the idea of introduced parameters and Euler-Maclaurin summation formula, a reverse extended Hardy-Hilbert inequality as well as the equivalent forms are given in Lemma 2 and Theorem 1. The equivalent statements of the best possible constant factor related to a few parameters and some particular cases are considered in Theorem 2 and Remark 1-2.

Lemma 2
We have the following reverse extended Hardy-Hilbert inequality: Proof In the same way as obtaining (7), for n ∈ N, we obtain the following inequality of the weight coefficient: By the reverse Hölder inequality (cf. [27]), we obtain Then, by (7) and (9), in view of 0 < p < 1, q < 0, we have (8).

Main results and some particular cases
Theorem 1 Inequality (8) is equivalent to the following inequalities: If the constant factor in (8) is the best possible, then so is the constant factor in (14) and (15).
If the constant factor in (8) is the best possible, then so is the constant factor in (14) and (15). Otherwise, by (16) (or (17)), we would reach a contradiction that the constant factor in (8) is not the best possible.