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On the Fekete-Szegö problem and the domain of convexity for a certain class of univalent functions. (English) Zbl 0741.30012
Let \(C_ \beta(\alpha)\) denote the class of regular functions \(f\) in the open unit disc \(U\) with \(f(z)=z+a_ 2z^ 2+a_ 3z^ 3+\ldots\) and Re\(\{e^{i\beta}(1-\alpha^ 2z^ 2)f'(z)\}>0\) for some \(0\leq\alpha\leq 1\) and \(\beta\) real with \(|\beta|<\pi/2\). The authors prove that \[ \max_{{f\in C_ \beta(\alpha)}\atop{0\leq\lambda\leq 1}}| a_ 3- \lambda a^ 2_ 2|={2\cos\beta+\alpha^ 2\over 3} \] and also describe the boundary curve of the set \[ E^ c_ \alpha=\{z\in U:\text{Re}\left(1+{zf''(z)\over f'(z)}\right)>0:f\in C_ 0(\alpha)\}. \]

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