Equivalent conditions and applications of a class of Hilbert-type integral inequalities involving multiple functions with quasi-homogeneous kernels

Abstract

Let $K(x_1,\dots ,x_n)$ satisfy $K(x_1,\dots ,t{x}_i,\dots ,x_n)=t^{\lambda {\lambda }_i}K(t^{-\frac{{\lambda }_i}{{\lambda }_1}}{x}_1,\dots ,x_i,\dots ,t^{-\frac{{\lambda }_i}{{\lambda }_n}}{x}_n)$ for $t>0$ . With this integral kernel, by using the method and technique of weight coefficients, the equivalent conditions and the best constant factors for the validity of Hilbert-type integral inequalities involving multiple functions are discussed. Finally, the applications of the integral inequalities are considered.

Keywords: Best constant factor; Equivalent conditions; Hilbert-type integral inequality; Quasi-homogeneous kernel.