Hilbert's integral inequality in whole plane with general homogeneous kernel

Abstract

The main objective of this paper is a study of some new generalizations of Hilbert's and Hardy-Hilbert's
type inequalities.
We build a new Hilbert's inequality with general homogeneous functions of degree $-2s,$ $s\geq 0$ in whole plane.
Also, we obtain the best
possible constants when the parameters satisfy appropriate conditions.