Equivalent Statements of Two Multidimensional Hilbert-Type Integral Inequalities with Parameters

Abstract

By means of the weight functions, the idea of introduced parameters and the transfer formulas, two multidimensional Hilbert-type integral inequalities with the general nonhomogeneous kernel as H(||x||αλ1||y||βλ2)(λ1,λ2≠0) are given, which are some extensions of the Hilbert-type integral inequalities in the two-dimensional case. Some equivalent conditions of the best value and several parameters related to the new inequalities are provided. Two corollaries regarding the kernel, represented as kλ(||x||αλ1,||y||βλ2)(λ1,λ2≠0), are given, and a few new inequalities for the particular parameters are obtained.

Keywords

• transfer formula
• multidimensional Hilbert-type inequality
• gamma function
• best possible constant factor