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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.

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