Let $A$ and $B$ be positive-definite matrices such that $A \le B.$

By matrix monotonicity of the root, this also implies that $A^{\alpha} \le B^{\alpha}$ for $\alpha \in [0,1].$

I am now curious under what assumptions we have

$$ A^{\alpha} B A^{\alpha} \le B^{1+2\alpha}$$ in the sense of operators?

Let $A$ and $B$ be positive-definite matrices such that $A \le B.$

By matrix monotonicity of the root, this also implies that $A^{\alpha} \le B^{\alpha}$ for $\alpha \in [0,1].$

I am now curious under what assumptions we have

$$ A^{\alpha} B A^{\alpha} \le B^{1+2\alpha}$$ in the sense of operators?

Edit: Notice that it does not seem to follow from Furuta's inequality, since we are changing the operator $A$.