  We establish a criterion for local boundedness and hence normality of a
family $\F$ of analytic functions on a domain $D$ in the complex plane whose
corresponding family of derivatives is locally bounded. Furthermore we
investigate the relation between domains of normality of a family $\F$ of
meromorphic functions and its corresponding Schwarzian derivative family. We
also establish some criterion for the Schwarzian derivative family of a family
$\F$ of analytic functions on a domain $D$ in the complex plane to be a normal
family.
