Abstract
By applying the weight functions and the idea of introduced parameters we give a new Hilbert-type integral inequality involving the upper limit functions and the beta and gamma functions. We consider equivalent statements of the best possible constant factor related to a few parameters. As applications, we obtain a corollary in the case of a nonhomogeneous kernel and some particular inequalities.
Highlights
If 0 < ∞ m=1 a2m < ∞ and ∞ n=1 b2n∞, we have the following discreteHilbert inequality with the best possible constant factor π ([1], Theorem 315):∞ ∞ ambn < π m+n a2m b2n
In 2016–2017, by applying the weight functions Hong [22, 23] considered some equivalent statements of the extensions of (1) and (2) with a few parameters
We consider the equivalent statements of the best possible constant factor related to a few parameters
Summary
Hilbert inequality with the best possible constant factor π ([1], Theorem 315):. ∞ ∞ ambn < π m+n a2m b2n. Hilbert inequality with the best possible constant factor π ([1], Theorem 315):. 0 0 x+y where the constant factor π is the best possible. The following half-discrete Hilbert-type inequality was provided in 1934 Considered an extension of (1) involving the partial sums. In 2016–2017, by applying the weight functions Hong [22, 23] considered some equivalent statements of the extensions of (1) and (2) with a few parameters. In this paper, following [21, 22], by the use of the weight functions and the idea of introduced parameters, we give a new Hilbert-type integral inequality with the kernel. We consider the equivalent statements of the best possible constant factor related to a few parameters. We obtain a corollary in the case of nonhomogeneous kernel and some particular inequalities
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