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Zbl 1038.30009
Dziok, J.; Srivastava, H. M.
Some subclasses of analytic functions with fixed argument of coefficients associated with the generalized hypergeometric function.
(English)
[J] Adv. Stud. Contemp. Math., Kyungshang 5, No. 2, 115-125 (2002). ISSN 1229-3067

Let $_qF_s(\alpha_1,\dots \alpha_q;\beta_1,\dots,\beta_s;z)$, $q \leq s+1$, $q,s=0, 1, 2,\dots$, $\alpha_j \in \Bbb{C}$, $\beta_j \in \Bbb{C}\setminus \{0, -1, -2\dots\}$ denote the generalized hypergeometric functions. Put $h(z)=z_qF_s$ and $H=h\star f$, where $f(z)=z+\sum_{n=2}^{\infty}a_n z^n$ is holomorphic in the unit disc $\vert z\vert < 1.$ The symbol $\star$ denotes the convolution or Hadamard product of two holomorphic functions.\par Under additional assumptions that $\alpha_j$ and $\beta_j$ are real and positive, the following class of functions $$T=T(q,s,A,B)= \left\{f(z)=z+ \sum_{n=2}^{\infty} a_n z^n:\frac{h \star f}{z}\prec\frac{1+Az}{1+Bz}\right\}$$ is defined. ($\prec$ means subordination and $0 \leq B \leq 1,\;\;-B \leq A < B$ ). The subclass $T_ \theta$ consisting of functions having the same argument of all coefficients is considered as well. \newline Coefficient estimates, some distortion theorems and radii of convexity and starlikeness are presented for the class $T$ and $T_\theta$. The results are sharp if $\theta=\pi.$
[Jan Szynal (Lublin)]
MSC 2000:
*30C45 Special classes of univalent and multivalent functions
30C50 Coefficient problems for univalent and multivalent functions
33C20 Generalized hypergeometric series

Keywords: generalized hypergeometric function; Hadamard product; subordination

Cited in: Zbl 1158.30313

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