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.


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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