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Meromorphic starlike univalent functions with alternating coefficients. (English) Zbl 0836.30006
Let \({\mathcal F}\) denote the family of all functions of the form \(f(z) = 1/z + \sum_{ m = 1}^\infty (-1)^{m - 1} a_m z^m\), \(a_m \geq 0\), which are regular in the punctured unit disc \(\Delta \backslash \{0\} = \{z : 0 < |z |< 1\}\) and satisfy the condition \(\text{Re} (D^{n + 1} f(z)/D^n f(z)) < 2 - \alpha\) for some \(n = 0,1, \dots\) and \(0 \leq \alpha < 1\), where \(D^n f(z)\) denotes the Hadamard product \(f(z)*(1/z + \sum_{m = 1}^\infty (m + 2)^n z^m)\). The authors obtain necessary and sufficient conditions, in terms of the coefficients, for functions \(f\) to belong to \({\mathcal F}\). Some other results proved in this paper are the distortion and growth theorems for functions in \({\mathcal F}\).
MSC:
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C50 Coefficient problems for univalent and multivalent functions of one complex variable
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