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On certain analytic functions associated with Ruscheweyh derivatives and bounded Mocanu variation. (English) Zbl 1155.30007
The authors study the class $$\mathcal A$$ of functions $$f$$ having the form $$f (z) = z + \sum^{\infty}_{m=2} a_m z^m$$ that are analytic in the unit disk $$E = \{z : |z| < 1\}$$ and the subclasses of functions in $$\mathcal A$$ that are univalent, close-to-convex, starlike, convex, of bounded radius rotation, of bounded Mocano variation, or Paatero bounded radius rotation.
The authors give a unified approach to the study via a huge new parametrized family of classes denoted by $$R_k (\alpha, a, \gamma)$$, where $$k \geq 2,\alpha \geq 2, a > 0$$, and $$0\leq \gamma < 1$$. The definition of $$R_k (\alpha, a,\gamma )$$ is complicated, but, for example:
$$R_k (0, 1, 0)$$ is the class of bounded radius rotation;
$$R_2 (\alpha, 1, 0)$$ is the class of alpha-starlike functions;
$$R_k (1, 1, 0)$$ is the class of Paatero bounded boundary rotation;
$$R_k (\alpha, 1, 0)$$ is the class of bounded Mocano variation;
$$R_2 (1, 1, 0)$$ is the class of convex functions; and
$$R_2 (0, 1, 0)$$ is the class of starlike functions.
The authors’ main results are inclusion relationships among the classes $$R_k (\alpha, a,\gamma )$$ and a crieterion for the univalence of the Ruscheweyh derivatives of functions in some of these classes.

##### 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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