Please use this identifier to cite or link to this item: http://hdl.handle.net/10077/4171

 Title: On the derivatives of a family of analytic functions Authors: Al-Kharsani, H. A.Al-Khal, R. A. Keywords: analytic functionsHadamard productpartial sumsextreme pointsconvex hull Issue Date: 2003 Publisher: Università degli Studi di Trieste. Dipartimento di Matematica e Informatica Citation: H.A. Al-Kharsani, R.A. Al-Khal, "On the derivatives of a family of analytic functions", in: Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics, 35 (2003), pp. 1-17. Series/Report no.: Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics35 (2003) Abstract: For $\beta< 1$, n = 0, 1, 2, . . ., and $-\pi <\alpha\leq\pi$, we let $M_n(\alpha,\beta)$ denote the family of functions $f(z) = z +\ldots$ that are analytic in the unit disk and satisfy there the condition $Re\{(D^n f)'+\frac{1+e^{i\alpha}}{2(n+1)}z(D^n f)''\}>\beta$, where $D^n f(z)$ is the Hadamard product or convolution of f with $z/(1 − z){n+1}$. We prove the inclusion relations $M_{n+1}(\alpha,\beta) \subset M_n(\alpha,\beta$, and $M_n(\alpha,\beta) < M_n(\pi,\beta), \beta < 1$. Extreme points, as well as integral and convolution characterizations, are found. This leads to coefficient bounds and other extremal properties. The special cases $M_0(\alpha,0)\equiv \mathcal{L}_\alpha$, $M_n(\pi,\beta)\equiv M_n(\beta)$ have previously been studied [16], [1]. URI: http://hdl.handle.net/10077/4171 ISSN: 0049-4704 MS Classification: 30C4530C50 Appears in Collections: Rendiconti dell' Istituto di matematica dell‘ Università di Trieste: an International Journal of Mathematics vol.35 (2003)

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