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Convex functions and functions with bounded turning. (English) Zbl 1211.30032

Summary: Let 𝒜 be the class of analytic functions in the unit disk 𝒰={z:|z|<1} that are normalized by f(0)=f ' (0)-1=0, and let -1B<A1 and -1D<C1. In this paper, the following generalizations of the class of convex functions and of the class of functions with bounded turning are studied:

K[A,B]=f 𝒜 : 1 + zf '' (z) f ' (z) 1+Az 1+Bz

and

R k [C,D]=f 𝒜 : f ' (z) 1/k 1+Cz 1+Cz,k1·

Conditions for K[A,B]R k [D,D] are given, together with several corollaries for different choices of A, B, C, D, and k,.

MSC:
30C45Special classes of univalent and multivalent functions