Could we bound $\|(\mathbf{X}+\mathbf{Y})^{-\frac{1}{2}}-\mathbf{X}^{-\frac{1}{2}}\|_{\mathrm{mav}}$ if we can bound $\|\mathbf{Y}\|_{\mathrm{mav}}$?

Question

Here $\mathbf{X}$ and $\mathbf{Y}$ are positive definite matrices, and $\|\cdot\|_{\mathrm{mav}}$ represents the maximum-absolute-value norm: $$ \|\mathbf{A}\|_{\mathrm{mav}} = \max\{|\mathbf{A}_{i j}| \mid i=1, \ldots, m, j=1, \ldots, n\}. $$