  A normalized analytic function f is shown to be univalent in the open unit
disk D if its second coefficient is sufficiently small and relates to its
Schwarzian derivative through a certain inequality. New criteria for analytic
functions to be in certain subclasses of functions are established in terms of
the Schwarzian derivatives and the second coefficients. These include obtaining
a sufficient condition for functions to be strongly $\alpha$-Bazilevi\^c of
order $\beta$.
