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On meromorphic starlike and convex functions. (English) Zbl 1013.30005

Let \(\Sigma\) denote the class of functions \(F\) of the form \(F(z)=\frac{1}{z}+\sum_{n=0}^{\infty}a_{n}z^{n}\), which is analytic and univalent in \(D=\{z:0<|z|<1\}\). A function \(F\in\Sigma\) denoted by \(MS^{*}(\alpha)\) is called meromorphic starlike of order \(\alpha(\alpha<1)\), if \(F(z)\neq 0\) in \(D\) and \(-\text{Re}\dfrac{zF'(z)}{F(z)}>\alpha\), \(z\in E\), \(E=\{z:|z|<1\}\). Similarly, a function \(F\in\Sigma\) denoted by \(MC(\alpha)\) is called meromorphic convex of order \(\alpha<1\) if \(F(z)\neq 0\) in \(D\) and \(-(1+\text{Re}\dfrac{zF''(z)}{F'(z)})>\alpha\), \(z\in E\). The author shows that the aforementioned functions do not hold the same relationship to the usual starlike univalent functions and convex univalent functions.

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C55 General theory of univalent and multivalent functions of one complex variable
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