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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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References:

[1] DOI: 10.1090/S0002-9939-1966-0188423-X
[2] Montel P., Leeons sur les Fonctions Univalentes ou Multivalentes (1933) · JFM 59.0346.14
[3] Owa S., I. Kyungpook Math. J. 18 pp 53– (1978)
[4] DOI: 10.4153/CJM-1987-054-3 · Zbl 0611.33007
[5] DOI: 10.1090/S0002-9939-1975-0367176-1
[6] Srivastava H. M., Comment. Math. Univ. St. Paul. 35 pp 175– (1986)
[7] Srivastava H. M., Nagoya Math. J. 106 pp 1– (1987) · Zbl 0607.30014
[8] Srivastava H. M., Univalent Functions, Fractional Calculus, and Their Applications (1989) · Zbl 0683.00012
[9] Srivastava H. M, Current Topics in Analytic Function Theory (1992) · Zbl 0976.00007
[10] DOI: 10.1016/0022-247X(92)90373-L · Zbl 0760.30006
[11] DOI: 10.1006/jmaa.1995.1197 · Zbl 0831.30008
[12] DOI: 10.1090/S0002-9947-1969-0232920-2
[13] DOI: 10.1137/0515057 · Zbl 0567.30009
[14] DOI: 10.1016/S0096-3003(98)10042-5 · Zbl 0937.30010
[15] Hohlov Yu. E., Izv. Vys?. Ucebn. Zaved. Matematika 10 pp 83– (1978)
[16] DOI: 10.1090/S0002-9939-1965-0178131-2
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