Inclusion criteria for subclasses of functions and Gronwall's inequality

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 α-Bazilevic of order β. Keywords and Phrases: Univalent functions, Bazilevic functions, Gronwall's inequality, Schwarzian derivative, Second coefficient.
1. Introduction Let A be the set of all normalized analytic functions f of the form f(z) = z + [∑.sub.k=2.sup.∞][a.sub.k][z.sup.k] defined in the open unit disk D: = {z [...]