On meromorphic starlike functions of reciprocal order \(\alpha\) (Q412002)

Let \(\sum\) denote the class of functions \(f\) of the form \[ f(z)=\frac{1}{z}+\sum^{\infty}_{k=0}a_{k}z^{k}, \] which are analytic in the punctured open unit disk \[ \mathbb{U}^{*}:=\{z:z \in \mathbb{C} \text{ and } 0<|z|<1\}=: \mathbb{U} \backslash \{0\}. \] A function \(f \in \sum\) is said to be in the class \(\mathbf{MS}^{*} (\alpha)\) of meromorphic starlike functions of order \(\alpha\) if it satisfies the inequality \[ Re \left( \frac{zf^{\prime}(z)}{f(z)} \right) < - \alpha \quad(0 \leq \alpha <1; z \in \mathbb{U}). \] As usual, let \(\mathbf{MS}^{*}(0) \equiv \mathbf{MS}^{*}.\) Furthermore, a function \(f \in \mathbf{MS}^{*}\) is said to be in the class \(\mathbf{NS}^{*} (\alpha)\) of meromorphic starlike of reciprocal order \(\alpha\) if and only if \[ Re \left( \frac{f(z)}{zf^{\prime}(z)} \right) < - \alpha \quad (0 \leq \alpha <1; z \in \mathbb{U}). \] In the present investigation, the authors give some sufficient conditions for functions to belong to the class \(\mathbf{NS}^{*} (\alpha)\), such as the following one: Theorem. If \(f \in \sum\) satisfies \[ \sum^{\infty}_{k=0} (1+k \alpha) |a_{k}| \leq \frac{1}{2} (1- |1-2\alpha|), \] then \(f \in \mathbf{NS}^{*}(\alpha)\), for \(0< \alpha <1\).