On starlike functions of order \(\lambda\in[{1\over 2},1)\) (Q2769222)
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scientific article; zbMATH DE number 1701120
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On starlike functions of order \(\lambda\in[{1\over 2},1)\) |
scientific article; zbMATH DE number 1701120 |
Statements
5 February 2002
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function starlike of order \(\alpha\)
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\(n\)-th partial sum of a function
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convex univalent function
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convolution (Hadamard product) of analytic functions
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stable functions
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Gegenbauer polynomials
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On starlike functions of order \(\lambda\in[{1\over 2},1)\) (English)
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Let \(zf\) be starlike of order \(\lambda \in [{1 \over 2}, 1)\), and denote by \(s_n(f,z)\) the \(n\)-th partial sum of the Taylor expansion of \(f\) about the origin. The authors then prove that NEWLINE\[NEWLINE {s_n(f,z) \over f(z)}\prec (1-z)^{2\lambda -2}, \quad n \in \mathbb N, NEWLINE\]NEWLINE where \(\prec\) denotes subordination. Applications to Gegenbauer polynomial sums are mentioned, and a new concept of ``stable'' functions is briefly discussed.
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