  A polynomial $p \in \mathbb{R}[z_1, \cdots, z_n]$ is called real stable if it
is non-vanishing whenever all the variables take values in the upper half
plane. A well known result of Elliott Lieb and Alan Sokal states that if $p$
and $q$ are $n$ variate real stable polynomials, then the polynomial
$q(\partial)p := q(\partial_1, \cdots, \partial_n)p$, is real stable as well.
In this paper, we prove analytical estimates on the locations on the zeroes of
the real stable polynomial $q(\partial)p$ in the case when both $p$ and $q$ are
multiaffine, an important special case, owing to connections to negative
dependance in discrete probability. As an application, we prove a general
estimate on the expected characteristic polynomials upon sampling from Strongly
Rayleigh distributions. We then use this to deduce results concerning two
classes of polynomials, mixed characteristic polynomials and mixed
determinantal polynomials, that are related to the Kadison-Singer problem.
