Let $f(a,b,c)=\det\begin{pmatrix}a &b\\ b& c\end{pmatrix}\in\mathbb{R}[a,b,c]$ be the determinant of a $2 \times 2$ real symmetric matrix. Let $f(x_i,y_i,z_i)\geq 0$, $x_i\geq 0$, $z_i\geq 0$ for $i\in\{1,2\}$.

(**Edit:** i.e. the matrix is positive semidefinite).

Then

$$ f(x_1+x_2,y_1+y_2,z_1+z_2)\geq 0 \tag{$\ast$} $$

(something that can be formulated as convexity of the cone of symmetric $2 \times 2$ positive semidefinite (p.s.d.) matrices).

Indeed, $(*)$ can be seen by viewing $f$ as the discriminant of the quadratic polynomial $$F(a,b,c)(t)=at^2+2bt+c \tag{**}$$ with $a\geq 0$, $c\geq 0$, observing that then $f(a,b,c)\geq 0$ iff $F(a,b,c)(t)\geq 0$ for all $t$, and seeing that $$F(x_1+x_2,y_1+y_2,z_1+z_2)(t)=F(x_1,y_1,z_1)(t)+F(x_2,y_2,z_2)(t)\geq 0$$

The question is whether there is a (more) direct proof of $(*)$, and what kinds of generalisations are known. E.g. I am interested in the situation where $a,b,c$ are multivariate polynomials, and under which conditions $f(a,b,c)$ is a sum of squares (s.o.s.) of polynomials---with potential applications to efficient s.o.s. decompositions.

**Edit:** As well, is there an analogue of $(**)$ for higher order positive semidefinite matrices?

hyperbolic polynomials, a notion due to Garding. $\endgroup$ – Denis Serre Jun 28 '17 at 11:19