Log convexity of various functions defined on the space of Hermitian matrices plays an important role in matrix analysis and probability theory.
Given $v \in \mathbb{C}^n$, $D$ a diagonal matrix with positive diagonal entries, $H$ a Hermitian matrix, and $k>0$ an integer, is it true that $$\log \left( \left| (e^{tH} D)^k \cdot v \right|^2 \right)$$ is convex as a function of $t$?
When $H$ and $D$ commute, this function is of the form $\log(\sum_i c_i e^{\lambda_i t})$ with $c_i>0$, which is convex. Some naive numerical simulations suggest that this is plausible is general, but I'm not familiar enough with the literature to track down where something like this might be discussed.
