Equivalent Property of a Hilbert-Type Integral Inequality Related to the Beta Function in the Whole Plane

Abstract

By means of the technique of real analysis and the weight functions, a few equivalent statements of a Hilbert-type integral inequality with the nonhomogeneous kernel in the whole plane are obtained. The constant factor related to the beta function is proved to be the best possible. As applications, the case of the homogeneous kernel, the operator expressions, and a few corollaries are considered.

Equivalent properties of a Mulholland-type inequality with a best possible constant factor and parameters

“…In 2016, by using the techniques of real analysis, Hong [20] considered some equivalent statements of the general form of (1) with the homogeneous kernel related to a few parameters and a best possible constant factor. Other similar results about the extended integral inequalities (3) were obtained in [21][22][23][24].…”

On a reverse extended Hardy–Hilbert’s inequality

“…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].…”

On a New Extended Hardy–Hilbert’s Inequality with Parameters

“…In 2016, by means of the technique of real analysis, Hong [20] considered some equivalent statements of the extensions of (1) with the best possible constant factor related to several parameters. For some similar works on the extensions of (3) and (4), we refer the reader to [21][22][23][24][25].…”