Let $U$ be a random matrix, supported on the positive-definite cone of matrices. We denote $\sqrt{U}$ to be the principal square root of $U$. That is, the unique positive-definite matrix such that $\sqrt{U}^2 = U$.

I'm interested in bounding the quantitiy $\text{Tr}({\mathbb{E}[\sqrt{U}]^2})$.

In the given answer here a solution is given for random variables which shows that $$\mathbb{E}[X]\left(1 - \frac{\mathbb{E}(X - \mathbb{E}[X])^2}{2\mathbb{E}[X]}\right)^2 \leq E[\sqrt{X}]^2$$.

However, the method used above is inapplicable for non-commutative matrices.

I'm wondering if there is any equivalent results known for matrices.