Let $A$ be real symmetric and $D$ shall contain the eigenvalues of $A$. I've learned that $\|A\|_{\text{max}}< \|D\|_{\text{max}}$, where $\|A\|_{\text{max}}$ means the Max norm.
I want to get a sharper bound $\|A\|_{\text{max}}$ by using the knwoledge about more than one eigenvalue, let say two. Ky-Fan norms seem appropriate, so I'm looking for something like $ \|A\|_{\text{max}}\not <\frac12\|D\|_2, $ where $\|D\|_2=\lambda_0+\lambda_1$ sums up the largest eigenvalues. Numerics showed that it doesn't hold, ven if I use absoulute values $|\lambda_0|+|\lambda_1|$.