A necessary condition for certain functions to preserve positive semi-definiteness on partitioned matrices

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.