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Differential subordinations and inequalities in the complex plane. (English) Zbl 0633.34005

Let f and F be analytic in the unit disc U. The function f is subordinate to F, written fF or f(z)F(z), if F is univalent, f(0)=F(0) and f(U)F(U). The authors deal with second order differential subordinations of the form (1)ψ(p(z),zp ' (z),z 2 p '' (z);z)h(z), where ψ : 3 ×U. 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 p(0)=0, then |zp ' (z)|+z 2 p '' (z)/p(z)|<1 implies that |p(z)|<1;

Theorem 7. If p is analytic in U with p(0)=1, and if Re[2p(z)-zp '' (z)/p ' (z)-1]>0, then Re p(z)>0.

Reviewer: N.V.Grigorenko

MSC:
34M99Differential equations in the complex domain