Abstract
In this article, by using weight functions, the idea of introducing parameters, the reverse extended Hardy–Hilbert integral inequality and the techniques of real analysis, a reverse Hardy–Hilbert-type integral inequality involving one derivative function and the beta function is obtained. The equivalent statements of the best possible constant factor related to several parameters are considered. The equivalent form, the cases of non-homogeneous kernel and some particular inequalities are also presented.
Highlights
Assuming that 0 < ∞ m=1 a2m < ∞ and ∞ n=1 b2n∞, we have the followingHilbert inequality with the best possible constant factor π:∞ ∞ ambn < π m+n a2m b2n
In 2016–2017, by applying the weight functions, Hong et al [23, 24] considered some equivalent statements of the extensions of (1) and (2) with several parameters
The equivalent statements of the best possible constant factor related to several parameters are considered
Summary
Hilbert inequality with the best possible constant factor π (cf. [1], Theorem 315):. ∞ ∞ ambn < π m+n a2m b2n. Hilbert inequality with the best possible constant factor π (cf [1], Theorem 315):. 0 0 x+y where the constant factor π is the best possible. The following half-discrete Hilbert-type inequality was presented in 1934 In 2016–2017, by applying the weight functions, Hong et al [23, 24] considered some equivalent statements of the extensions of (1) and (2) with several parameters. In this paper, following [21, 23], by the use of weight functions, the idea of introducing parameters, the reverse extension of (1) and the technique of real analysis, a reverse. The equivalent statements of the best possible constant factor related to several parameters are considered. The equivalent form, the cases of non-homogeneous kernel and a few particular inequalities are obtained
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