Hadamard-type inequalities for hyperbolic polynomials

Let $P$ be a homogeneous, real, symmetric, hyperbolic polynomial of degree $k$ on $F?$. Let $AF?M_n(F?)$. If the eigenvalue vector $F? (A)F?\Gamma(P)$, then $(a_{11},\ldots,a_{nn})F?\Gamma(P)$ and

$$P(a_{11},\ldots,a_{nn})\geq P(\lambda(A)).$$

Hadamard-type inequality for hyperbolic polynomials. The claim extends the diagonal-versus-eigenvalue inequality from elementary symmetric polynomials to homogeneous, real, symmetric, hyperbolic polynomials. Its resolution is not indicated in the supplied text.