Hilbert-type inequalities with a product-type homogeneous kernel and Schur polynomials

The main objective of this paper is to prove Hilbert-type and Hardy–Hilbert-type inequalities with a product-type homogeneous kernel, thus generalizing a result obtained in [Z. Xie, Z. Zheng, A Hilbert-type integral inequality whose kernel is a homogeneous form of degree −3, J. Math. Anal. Appl. 339 (2008) 324–331]. In some cases the best possible constants obtained in these inequalities are expressed using the Schur polynomials.