For real 1)$"> α ( α > 1 ) , we introduce subclasses M ( α ) and N ( α ) of analytic functions f ( z ) with f ( 0 ) = 0 and f ′ ( 0 ) = 1 in U . The object of the present paper is to consider the coefficient inequalities for functions f ( z ) to be in the classes M ( α ) and N ( α ) . Further, the bounds of α for functions f ( z ) to be starlike in U are considered.