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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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