The classical Cordes inequality states the following: suppose that $\|\cdot\|$ is the usual matrix norm, $0 < \alpha \leq 1$, and $A, B$ are $n \times n$ positive semidefinite Hermitian matrices. Then we have $$ \|A^\alpha B^\alpha \| \leq \|A B \|^\alpha.$$ See for example Furuta's "Norm Inequalities Equivalent to Löwner-Heinz Theorem" (1991).
In what I'm doing, I'm OK with upper bound $1$ being replaced with $C = C_{n, \alpha}$ and here's a quick proof of this with $C = C_n$. Namely By the classical Hölder-McCarthy inequality we have \begin{align*} \|A^\alpha B^\alpha\| & \approx \sum_{j = 1}^n |A^\alpha B^\alpha \vec{e}_j| = \sum_{j = 1}^n \lambda_j ^\alpha |A^\alpha \vec{e}_j| \leq \sum_{j = 1}^n \lambda_j ^\alpha | A \vec{e}_j| ^\alpha \\ & = \sum_{j = 1}^n |A \lambda_j \vec{e}_j| ^\alpha = \sum_{j = 1}^n |A B\vec{e}_j| ^\alpha \approx \|AB\|^\alpha \end{align*}
What (if anything) can be said about comparing $|A^\alpha B^\alpha \vec{e}|$ to $|AB \vec{e}|^\alpha$ if $\vec{e}$ is a unit vector and $0 < \alpha \leq 1$? In particular, does $$ |A^\alpha B^\alpha \vec{e}| \leq C_{n, \alpha} |A B \vec{e}|^\alpha$$ hold? Can the easy proof above be modified to prove this?
P.S. As for why I'm interested, these types of inequalities are useful for proving Matrix weighted norm inequalities of various types, and I'm interested in applying them towards understanding matrix A${}_{p, \infty}$ better (particularly going beyond the paper https://arxiv.org/abs/2311.05974 by T. Hytönen et. al. for example.)
