Let n ≥ 2. We prove a condition on $$f\in {C}^{2}\left({\hbox{ R }}_{+},\hbox{ R }\right)$$ for the convexity of f det on PSym(n), namely that f det is convex on PSym(n) if and only if

\[ {f}^{\prime\prime }\left(s\right)+\frac{n-1}{ns}\cdot {f}^{\prime }\left(s\right)\ge 0\phantom{\rule{0.25em}{0ex}}\hbox{ and }\phantom{\rule{0.25em}{0ex}}{f}^{\prime }\left(s\right)\le 0\phantom{\rule{0.25em}{0ex}}\forall \phantom{\rule{0.25em}{0ex}}s\in {\hbox{ R }}_{+}. \]

This generalizes the observation that C ↦ –ln det C is convex as a function of C.