A Hilbert-Type Integral Inequality in the Whole Plane Related to the Hypergeometric Function and the Beta Function

摘要

A new Hilbert-type integral inequality in the whole plane with thenon-homogeneous kernel and parameters is given. The constant factor related tothe hypergeometric function and the beta function is proved to be the bestpossible. As applications, equivalent forms, the reverses, some particularexamples, two kinds of Hardy-type inequalities, and operator expressions areconsidered.