Let f and F be analytic in the unit disc U. The function f is subordinate to F, written or f(z), if F is univalent, and f(U). The authors deal with second order differential subordinations of the form where : . They generalize their previous results [see, Mich. Math. J. 28, 157-171 (1981; Zbl 0439.30015)] on the case (1). With help from this generalization they prove some new inequalities, for example:
Theorem 6. If p is analytic in U with , then implies that
Theorem 7. If p is analytic in U with , and if then Re p(z).