For finite dimensional spaces, all norms are equivalent, i.e. there exist constants say $A,B$ such that for all matrices from the $\mathbf M \in R^{d\times d}$ (let $d$ be a fixed positive integer) such that
$A \|\mathbf M\|_2 \leq \|\mathbf M \|_F \leq B\|\mathbf M\|_2\text{,} $
where $\|\cdot\|_2$ denotes the spectral norm and $\|\cdot\|_F$ (edit: i forgot the second part...) denotes the Frobenius norm.
My question is now, whether you know anything specific about $A$ and $B$, for example whether there exists a analytical expression for let's say $B$.