  If $f$ is a symmetric complex-valued function on the $m$-fold Cartesian
product of the set of non-negative reals and $A$ is a positive semi-definite
$m\times m$ matrix with eigenvalues $\lambda_j$, we set
$f(A):=f(\lambda_1,\dotsc,\lambda_m)$. It is shown that if
$[f(A_{\alpha\beta})]$ is positive semi-definite whenever $[A_{\alpha\beta}]$
is a positive semi-definite matrix with positive semi-definite entries
$A_{\alpha\beta}$, then $f$ has a power series expansion with positive
coefficients.
