Lets say $A$ and $B$ are two Hermitian matrices with positive eigenvalues. 

Let say $k>0$ is a integer.

Let $P=(P_1,P_2,\cdots,P_k)$ be an arbitrary sequence of $k$ $A$ and $B$s in any given order.

Is there a trace inequality in form of

$tr\,\prod^{k}_{i=1} P_i \leq $  $tr\,A^k B^k$ ?