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Certain subclasses of analytic functions associated with the generalized hypergeometric function. (English) Zbl 1040.30003
For two holomorphic functions in the unit disk $$| z| < 1,$$ $$f(z)=z+\sum_{n=2}^{\infty}a_n z^n$$ and $$g(z)=z+\sum_{n=2}^{\infty}b_n z^n$$ the Hadamard product (or convolution ) is defined as $$(f \star g)(z)=z+\sum_{n=2}^{\infty}a_n b_n z^n.$$ Let $$_qF_s(\alpha_1,\ldots\,\alpha_q; \beta_1,\ldots,\beta_s;z)$$ denote the generalized hypergeometric function with $$q \leq s+1$$, $$q,s=0, 1, 2,\ldots$$, $$\alpha_j \in \mathbb{C}$$, $$\beta_j \in \mathbb{C}\setminus \{0, -1, -2\ldots\}$$. Very special subclasses of holomorphic functions $$f(z)=a_1 z-\sum_{n=2}^{\infty}a_n z^n$$, $$a_n \geq 0$$, $$n\geq2$$, $$| z| <1$$ defined by convolution of $$f$$ with $$z_qF_s$$ and subordinate to $$\frac{1+Az}{1+Bz}$$, $$0 \leq B \leq 1$$, $$-B \leq A \leq B$$ are studied.
(They satisfy moreover the condition $$f(\rho)=\rho$$ or $$f'(\rho)=1$$, $$0<\rho<1$$.) The necessary and sufficient conditions in the terms of coefficients for $$f$$ to be in the class under consideration are determined. Some distortion theorems and the radii of convexity and starlikeness are found.

##### MSC:
 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) 33C20 Generalized hypergeometric series, $${}_pF_q$$
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