# Matrix convexity of -log

Is $-\log$ a matrix convex function? That is, taking the function $\log:(0,\infty)\rightarrow \mathbb{R}$ is the matrix inequality $$\log\left((1-t)A+tB \right)\geq (1-t)\log A+ t \log B$$ satisfied for all matrices $A$ and $B$ with positive eigenvalues and $t\in[0,1]$?

It turns out that the functions $g_\alpha:(0,\infty)\rightarrow\mathbb{R}$ of the form $$g_\alpha(x)=\frac{x^{1-\alpha}-1}{1-\alpha}$$ are matrix concave for $\alpha\in(0,1)\cup(1,2)$ (see http://www.scholarpedia.org/article/Matrix_and_Operator_Trace_Inequalities) and $\log(x)$ is the limit of these as $\alpha\rightarrow 1$. So $\log$ is matrix concave.