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 $ $tr\,A^k B^k$ ?

Let $A$ and $B$ be two Hermitian matrices with positive eigenvalues. Let $k>0$ be a integer. Let $P=(P_1,P_2,\dots,P_{2k})$ be a sequence of $k$ $A$s and $k$ $B$s in any given order.

Do we have ${\rm tr}\,\prod^{2k}_{i=1} P_i \leq {\rm tr}\,A^k B^k$ ?

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

Let say $k>0$ is a integer.

Let $P$ be the arbitrary sequence of $k$ $A$s and $k$ $B$s in any given order.

Is there a trace inequality like:

tr $\Pi P \leq $ tr $A^k B^k$ ?

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 $ $tr\,A^k B^k$ ?