  A polynomial $p\in\mathbb{R}[z_1,\dots,z_n]$ is real stable if it has no
roots in the upper-half complex plane. Gurvits's permanent inequality gives a
lower bound on the coefficient of the $z_1z_2\dots z_n$ monomial of a real
stable polynomial $p$ with nonnegative coefficients. This fundamental
inequality has been used to attack several counting and optimization problems.
  Here, we study a more general question: Given a stable multilinear polynomial
$p$ with nonnegative coefficients and a set of monomials $S$, we show that if
the polynomial obtained by summing up all monomials in $S$ is real stable, then
we can lowerbound the sum of coefficients of monomials of $p$ that are in $S$.
We also prove generalizations of this theorem to (real stable) polynomials that
are not multilinear. We use our theorem to give a new proof of Schrijver's
inequality on the number of perfect matchings of a regular bipartite graph,
generalize a recent result of Nikolov and Singh, and give deterministic
polynomial time approximation algorithms for several counting problems.
