  In this paper we give some sufficient conditions for the nonnegativity of
immanants of square submatrices of Catalan-Stieltjes matrices and their
corresponding Hankel matrices. To obtain these sufficient conditions, we
construct new planar networks with a recursive nature for Catalan-Stieltjes
matrices. As applications, we provide a unified way to produce inequalities for
many combinatorial polynomials, such as the Eulerian polynomials, Schr\"{o}der
polynomials and Narayana polynomials.
