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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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