On multivariate Newton-like inequalities

We study multivariate entire functions and polynomials with non-negative coefficients. A class of Strongly Log-Concave entire functions, generalizing Minkowski volume polynomials, is introduced: an entire function f in m variables is called 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 (of propagation) on the standard univariate (or homogeneous bivariate) Newton Inequalities. We prove analogues of the Newton Inequalities in the multivariate Strongly Log-Concave case. One of the corollaries of our new Newton-like inequalities is the fact that the support supp(f) of a Strongly Log-Concave entire function f is pseudo-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 van der Waerden-Egorychev-Falikman inequality for the permanent of doubly-stochastic matrices. A few open questions are posed in the final section.