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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 $\vert z\vert < 1,$ $f(z)=z+\sum_{n=2}^{\infty}a_n z^n$ 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 $q \leq s+1$, $q,s=0, 1, 2,\ldots$, $\alpha_j \in \Bbb{C}$, $\beta_j \in \Bbb{C}\setminus \{0, -1, -2\ldots\}$. Very special subclasses of holomorphic functions $f(z)=a_1 z-\sum_{n=2}^{\infty}a_n z^n$, $a_n \geq 0$, $n\geq2$, $\vert z\vert <1$ 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.

30C45Special classes of univalent and multivalent functions
33C20Generalized hypergeometric series, ${}_pF_q$
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