Let us suppose that $A_{n}$ and $B_n$ are sequences of positive definite matrices satisfying

$c\leq \lambda_{\min}(A_n)\leq \lambda_{\max}(A_n)\leq C$

and

$c\leq \lambda_{\min}(B_n)\leq \lambda_{\max}(B_n)\leq C$

where $\lambda_{\min}$ and $\lambda_{\max}$ are minimal and maximal eigenvalues. Then, what's the relationship between $||A_n-B_n||_2$ and $||A_n^2-B_n^2||_2$? Is it true that $||A_n-B_n||_2 \leq \mathrm{constant}*||A_n^2-B_n^2||_2$, where the $\mathrm{L}_2$ norm of a matrix $M$ is the maximum of the absolute value of its minimal eigenvalue and the absolute value of its maximal eigenvalue.