I suspect that the following matrix inequality is well known, but I can't find a reference or proof:

Given $n \times n$ symmetric matrices $A,B$ such that $I_n \leq A,B$, is the following true? $${Tr}\big[\big( \log A^{\frac{1}{2}} B A^{\frac{1}{2}}\big)^p\big] \geq {Tr}\big[\big( \log B \big)^p\big] + {Tr}\big[\big( \log A\big)^p\big].$$

For clarity, for a continuous function $f$ and symmetric matrix $C$, $f(C)$ is simply equal to $U {diag}(f(\lambda_1),...,f(\lambda_n))U^t$, where $U$ is a unitary matrix diagonalizing $C$, i.e., $C = U diag(\lambda_1,...,\lambda_n)U^t$.

UPDATE: after the discussion with amakelov below, I edited the statement to be more aesthetic.