I have a covariance matrix $V=(v_{ij})$ and construct a correlation matrix $C$ with entries $c_{ij}=\frac{v_{ij}}{\sqrt{v_{ii}v_{jj}}}$. The matrices $V$ and $C$ are positive definite, so I know that the eigenvalues of $V$ and $C$ are real and positive. I am looking for a lower bound on $C$'s smallest eigenvalue and an upper bound on $C$'s largest eigenvalue.
My first idea was to make use of the bound on elements of $C$. I know that all diagonal elements of $C$ are $c_{ii}=1$ and that all non-diagonal elements $c_{ij}\in[-1,1]$. Using the Gershgorin circle theorem, all eigenvalues of $C$ lie in an interval $[1-n,1+n]$, where $n$ is the number of rows and columns in $C$.
I expect that one can do much better than this, perhaps by including a bit of information about the spectrum of $V$. For example, if the support of the spectrum of $V$ is a subset of an interval $[a,b]$, can we deduce bounds for the support of $C$ from this?