  We study multivariate entire functions and polynomials with non-negative
coefficients. A class of {\bf Strongly Log-Concave} entire functions,
generalizing {\it Minkowski} volume polynomials, is introduced: an entire
function $f$ in $m$ variables is called {\bf Strongly Log-Concave} if the
function $(\partial x_1)^{c_1}...(\partial x_m)^{c_m} f$ is either zero or
$\log((\partial x_1)^{c_1}...(\partial x_m)^{c_m} f)$ is concave on
$R_{+}^{m}$. We start with yet another point of view (via {\it propagation}) on
the standard univarite (or homogeneous bivariate) {\bf Newton Inequlities}. We
prove analogues of (univariate) {\bf Newton Inequlities} in the (multivariate)
{\bf Strongly Log-Concave} case. One of the corollaries of our new Newton(like)
inequalities is the fact that the support $supp(f)$ of a {\bf Strongly
Log-Concave} entire function $f$ is discretely convex ($D$-convex in our
notation). The proofs are based on a natural convex relaxation of the
derivatives $Der_{f}(r_1,...,r_m)$ of $f$ at zero and on the lower bounds on
$Der_{f}(r_1,...,r_m)$, which generalize the {\bf Van Der
Waerden-Falikman-Egorychev} inequality for the permanent of doubly-stochastic
matrices. A few open questions are posed in the final section.
