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Analytic functions of bounded radius rotation with respect to symmetrical points. (English) Zbl 0843.30017
The author introduces a class $R_k (s)$ of analytic functions $f(z) = z + a_2 z^2 + \cdots$ in the unit disc of the complex plane satisfying $$\int^{2 \pi}_0 \left |\text{Re} \left\{ {2zf' (z) \over f(z) - f( - z)} \right\} \right |d \theta \le k \pi$$ where $z = re^{i \theta}$ and $k \ge 2$. Representation theorems, covering theorems, distortion theorems, coefficient estimates are obtained. Integral operators are defined on this space and their images studied. Upper bounds for the length of the image of the circle $|z |= r$ $(0 < r < 1)$ under any $f \in R_k (s)$ are also obtained.

30C45Special classes of univalent and multivalent functions