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On some classes of close-to-convex functions. (English. Polish, Russian summaries) Zbl 0721.30008

The author introduces a class of analytic functions \(C(\alpha\)) which coincides with the class of functions convex in the direction of the imaginary axis when \(\alpha =1\), \(f\in C(\alpha)\) if \[ Re\{(1-\alpha^ 2z^ 2)f'(z)\}>0 \] in the open unit disc for some \(0\leq \alpha \leq 1\). It is proved that \(C(\alpha\)) consists of univalent functions and \(C(\alpha)\cap C(\beta)\neq \emptyset\) for \(\alpha,\beta\in [0,1]\). Coefficient estimates, distortion theorems and covering theorem are obtained in the canonical way.

MSC:

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