A homogeneous real polynomial, and showed various ways of constructing new hyperbolic
polynomials. We present a powerful new such construction, and use it to generalize
Gårding's result to arbitrary symmetric functions of the roots. Many classical and recent
inequalities follow easily. We develop various convex-analytic tools for such symmetric
functions, of interest in interior-point methods for optimization problems over related cones.

A homogeneous polynomial p (x) is hyperbolic with respect to a given vector d if the real
polynomial t 7! p (x+ td) has all real roots for all vectors x. We show that any symmetric
convex function of these roots is a convex function of x, generalizing a fundamental result of
G arding. Consequently we are able to prove a number of deep results about hyperbolic
polynomials with ease. In particular, our result subsumes von Neumann's characterization of
unitarily invariant matrix norms, and Davis's characterization of convex functions of the …