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.


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