I am trying to understand if there is a relation between the two problems
$\max_{\text{tr}\left(\boldsymbol{S}^{H}\boldsymbol{S}\right)=1}\text{tr}\left(\boldsymbol{S}^{H}\boldsymbol{A}\boldsymbol{S}\left(\boldsymbol{S}^{H}\boldsymbol{B}\boldsymbol{S}\right)^{-1}\right)$
and
$\max_{\text{tr}\left(\boldsymbol{S}^{H}\boldsymbol{S}\right)=1}\frac{\text{tr}\left(\boldsymbol{S}^{H}\boldsymbol{A}\boldsymbol{S}\right)}{\text{tr}\left(\boldsymbol{S}^{H}\boldsymbol{B}\boldsymbol{S}\right)}$
is one an upper\lower bound for the other?