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

$$\|A_n-B_n\|_2 \leq \text{constant} \cdot \|A_n^2-B_n^2\|_2$$

where the $2$-norm of a matrix $M$ is the maximum of the absolute value of its minimal eigenvalue and the absolute value of its maximal eigenvalue.