On multivariate Newton-like inequalities

Understand

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.