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On the convex combination of the Dziok-Srivastava operator. (English) Zbl 1124.33007

In 1999 Dziok and Srivastava have introduced the following operator \[ H_p(\alpha_1,\dots,\alpha_q; \beta_1,\dots,\beta_s): f(z) \mapsto h_p(\alpha_1,\dots,\alpha_q; \beta_1,\dots,\beta_s; z)\ast f(z) \] with \(h_p(\alpha_1,\dots,\alpha_q; \beta_1,\dots,\beta_s; z) = z^p\cdot _{q}F_{s}(\alpha_1,\ldots,\alpha_q; \beta_1,\dots,\beta_s; z)\) and \(_{q}F_{s}\) being the generalized hypergeometric function. This operator is known to be acting on the subspace \({\mathcal A}_p^k\) of those analytic in the unit disc functions having the following representation \[ f(z) = z^p + \sum_{n=k}^{\infty} a_n z^n. \] Convex combinations of the Dziok-Srivastava operators are introduced and studied in the paper. They are proved to be acting on the subclasses \(V_p^k(s; A, B; t)\) of those analytic functions having the form \[ f(z) = z^p - \sum_{n=k}^{\infty} a_n z^n \quad (a_n \geq 0) \] and satisfying the following condition \[ (1 - t) \frac{H_{p,s}(\alpha_1) f(z)}{z^p} + t \frac{H_{p,s}(\alpha_1 + 1) f(z)}{z^p} \prec \frac{1 + A z}{1 + B z}, \] where \(H_{p,s} = H_p(\alpha_1,\dots,\alpha_{s+1}; \beta_1,\dots,\beta_s)\). The set of the extreme points of these classes are characterized.

MSC:

33C20 Generalized hypergeometric series, \({}_pF_q\)
30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination
46B22 Radon-Nikodým, Kreĭn-Milman and related properties
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