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Hadamard product of certain starlike functions. (English) Zbl 0585.30013

For \(0\leq \alpha <1\) and \(0<\beta \leq 1\), let \(S^*(\alpha,\beta)\) denote the class of functions of the form \(f(z)=z- \sum^{\infty}_{n=2}a_ nz^ n\) where \(a_ n\geq 0\), f is analytic in \(\{\) \(z: | z| <1\}\) and \(| \frac{zf'(z)/f(z)- 1}{zf'(z)/f(z)+(1-2\alpha)}| <\beta\) for \(| z| <1\). Also, let \(C^*(\alpha,\beta)\) denote the class of functions f for which \(zf'(z)\in S^*(\alpha,\beta).\)
The author proves that if \(f\in S^*(\alpha_ 1,\beta_ 1),\quad g\in S^*(\alpha_ 2,\beta_ 2)\) and h is the Hadamard product of f and g, then \(h\in C^*(\alpha_ 1,\beta_ 1)\cap C^*(\alpha_ 2,\beta_ 2).\) The author also points out that this improves a result due to S. Owa [Math. Jap. 27, 409-416 (1982; Zbl 0497.30015)], and this follows primarily because \(C^*(\alpha,\beta)\subset S^*(\alpha,\beta).\)
Reviewer: Th.H.MacGregor

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)

Citations:

Zbl 0497.30015
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References:

[1] Gupta, V. P.; Jain, P. K., Certain classes of univalent functions with negative coefficients, Bull. Austral. Math. Soc., 14, 409-416 (1976) · Zbl 0323.30016
[2] Owa, S., On the classes of univalent functions with negative coefficients, Math. Japon., 27, 4, 409-416 (1982) · Zbl 0497.30015
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