On radius of univalence of convex combination of analytic functions. I (Q788859)

Let \(S,K,S^*\) and C denote the classes of analytic functions in the unit disc E which are univalent, close-to-convex, starlike and convex respectively. A function, analytic in E, is called quasi-convex if there exists a convex function g such that \(Re\frac{(zf'(z))'}{g'(z)} > 0\). The class of quasi-convex functions are denoted by \(C^*\). A function \(f\) analytic in \(E\), is called \(\alpha\)-quasi-convex if there exists a convex function \(g\) such that for \(\alpha\) real positive \[ \text{Re} \left[ (1- \alpha) \frac{f'(z)}{g'(z)} + \alpha \frac{(zf'(z))'}{g'(z)} \right] > 0. \] In this paper the mapping properties of \[ f(z) = (1-\alpha) F(z) + \alpha z F'(z), \quad \alpha>0, \] where F belongs to different subclasses of univalent functions are studied (as convex, starlike, quasi-convex and close-to-convex). It is established the radius \(r_ 0\) for which f is the same class with F. Related results to this one can also find in \textit{P. N. Chichra} [Proc. Am. Math. Soc. 62, 37-43 (1977; Zbl 0355.30013)], \textit{N. N. Pascu} [Romanian-Finnish seminar on complex analysis, Proc., Bucharest 1976, Lect. Notes Math. 743, 331-335 (1979; Zbl 0421.30009)] and \textit{N. Pascu, V. Podaru} and \textit{P. Rotaru [Bul. Univ. Brasov, Ser. C 19, 41-46 (1977; Zbl 0435.30009)].}