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On certain classes of convex functions. (English) Zbl 1279.30027

Summary: For real numbers \(\alpha\) and \(\beta\) such that \(0 \leq \alpha < 1 < \beta\), we denote by \(\mathcal K(\alpha, \beta)\) the class of normalized analytic functions which satisfy the following two sided-inequality: \[ \alpha < \mathrm{Re}\{1 + (zf''(z)/f'(z))\} < \beta \] where \(z \in \mathbb U\) and \(\mathbb U\) denotes the open unit disk. We find some relationships involving functions in the class \(\mathcal K(\alpha, \beta)\). We also estimate the bounds of coefficients and solve the Fekete-Szegő problem for functions in this class. Furthermore, we investigate the bounds of initial coefficients of inverse functions or bi-univalent functions.

MSC:

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

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