Convolution and differential subordination. (English) Zbl 0683.30023

Let A denote the class of functions analytic in \(U=\{z:\) \(| z| <1\}\) satisfying \(f(0)=(0)=f'(0)-1\). For a fixed \(g\in A\) denote by \(S_ g(h)\) the class of all functions \(f\in A\) such that g*f(z)\(\neq 0\) and \(z(g*f)'(z)/(g*f)(z)\prec h(z)\), for \(z\in U\), where h is any convex analytic univalent function on U with \(h(0)=1\) and Re h(z)\(>0\). Properties of the class \(S_ g(h)\) are in the main studied in the paper using the method of differential subordination. Certain related classes are also introduced and studied. Earlier results due to the reviewer occur as special cases.
Reviewer: K.S.Padmanabhan


30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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