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Applications of Ruscheweyh derivatives and Hadamard product to analytic functions. (English) Zbl 0964.30006
Denote by \(H\) the class of normalized functions \(f(z) = z + \sum_{m=2}^{\infty} a_{m}(f) z^{m}\) analytic in the unit disc. Given \(A, B,\) \(-1 \leq A < B \leq 1,\) and two functions \(\phi, \psi \in H\) such that \(0 \leq a_{m}(\psi) \leq a_{m}(\phi),\) \(m \geq 2,\) define the subclass \[ E_{n} (\phi, \psi; A, B) = \left\{f \in H: \frac{D^{n+1} (f* \phi)(z)}{D^{n} (f* \psi)(z)} \right. \left. \prec \frac{1+Az}{1+Bz} \right\}, \] where \(D^{n}h(z)= z (z^{n-1}h(z))^{(n)}/ n!,\) \(n \geq 0,\) is the \(n\)-th Ruscheweyh derivative; \(*\) and \(\prec\) stand for the Hadamard product and subordination, respectively. Set also \(E_{n} [\phi, \psi; A, B] = \{f \in E_{n} (\phi, \psi; A, B): a_{m} \leq 0, m \geq 2 \}.\)
Coefficient estimates, extreme points, distortion theorems and radii of starlikeness and convexity are found for \(E_{n} [\phi, \psi; A, B]\). The paper is concluded by showing that the quasi-Hadamard product of several factors from such classes with particular \(\phi\) and \(\psi\) belongs to certain generalization of it.

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