DIFFERENTIAL SUBORDINATION AND BAZILEVIC-FUNCTIONS

Let M(z) = z(n) +..., N(z) = z(n) + ... be analytic in the unit disc Delta and let lambda(z) = N(z)/zN'(z). The classical result of Sakaguchi-Libera shows that Re(M'(z)/N'(z)) > 0 implies Re(M(z)/N(z)) > 0 in Delta whenever Re(lambda(z)) > 0 in Delta. This can de expressed in terms of differential subordination as follows: for any p analytic in Delta, with p(0) = 1, p(z) + lambda(z)zp'(z) < 1 + z/1 - z implies p(z) <1 + z/1 - z, for Re lambda(z) > 0, z is an element of Delta. In this paper we determine different type of general conditions on lambda(z), h(z) and phi(z) for which one has p(z) + lambda(z) zp'(z) < h(z) implies p(z) < phi(z) < h(z), z is an element of Delta. Then we apply the above implication to obtain new theorems for some classes of normalized analytic functions. In particular we give a sufficient condition for an analytic function to be starlike in Delta.