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The set of starlikeness and some radius problem in certain classes of regular functions. (English) Zbl 0861.30012

Let \(\widetilde H_\alpha\), \(\alpha \in \langle 0,1 \rangle\), denote the class of all functions of the form \(f(z)= z+ a_2z^2+ \dots\) which are regular in the disc \(U= \{z\in C: |z|<1\}\) and satisfy the condition \(\text{Re}\{(1- \alpha^2z^2) f(z)/z\} > 0\), \(z\in U\). Of course, the class \(H_1 \subset \widetilde H_1\) of functions with real coefficients is identical with the known class \(T_R\) of typically-real functions of W. Rogosiński [Math. Z. 35, 93-121 (1932; Zbl 0003.39303)]. In [Demonstratio Math. 27, No. 2, 521-536 (1994; Zbl 0815.30012)] the authors introduced and examined the basic properties of functions of the classes \(\widetilde H_\alpha\). In this paper the set and the radius of starlikeness of the classes \(\widetilde H_\alpha\) are described. Some radius problem connected with the reciprocal dependence of the classes under consideration with respect to the parameter \(\alpha\) is also examined.

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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