The max norm is the elementwise norm with $p = \infty$: $ \|A\|_{\text{max}} = \max \{|a_{ij}|\}. $ This norm is not sub-multiplicative.
Let $A$ be real symmetric and $D$ shall contain the eigenvalues of $A$. How are $\|A\|_{\text{max}}$ and $\|D\|_{\text{max}}$ related?
Some examples seem to indicate that $\|A\|_{\text{max}}< \|D\|_{\text{max}}$. Is that all that one can say?