One of my professor claims that $\log f(X)$ concave implies that $\log(f(X^{-1}))$ convex where $X$ is symmetric positive definite matrix. $\log(f(X))$ is a function defined on symmetric positive definite matrix.
When I ask him why, he said if you do not believe it, give me a counterexample. So I wonder is it true or any counterexample to this claim.
I only know how to check a composition function convex by computing its Hessian. But what would be the Hessian in this case? I do not know how to compute.
Here is your counterexample.
Take $n = 1$ and the function $f(x) = e^{-x^2}$ on the positive real axis. Then $\log f(x) = -x^2$ which is a concave function, and $\log f(x^{-1}) = -x^{-2}$ which is also a concave function.
Don't believe everything your professor says, check it for yourself :)