Kumar, Vinod Hadamard product of certain starlike functions. (English) Zbl 0585.30013 J. Math. Anal. Appl. 110, 425-428 (1985). 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 Cited in 2 ReviewsCited in 9 Documents MSC: 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) Keywords:starlike functions; Hadamard product Citations:Zbl 0497.30015 PDFBibTeX XMLCite \textit{V. Kumar}, J. Math. Anal. Appl. 110, 425--428 (1985; Zbl 0585.30013) Full Text: DOI 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.