One criterion for meromorphic univalent functions (Q2739283)

Let \(\Sigma(n)\) be the class of functions \(f(z)=\frac{1}{z}+a_nz^n+\dots\) (\(n\in {\mathbb N}\cup\{0\}\)), which are analytic in the punctured disk \(D=\{z: 0<|z|<1\}\). Also let \(E=\{z: |z|<1\}\) be the unit disk. In this paper the author obtains the following sufficient criterion for meromorphic univalent functions. NEWLINENEWLINENEWLINETheorem. Let \(f(z)=\frac{1}{z}+a_nz^n+\dots\in \Sigma(n)\) (\(n\in {\mathbb N}\cup \{0\}\)) with \(zf(z)\neq 0\) for \(0<|z|<1\). If NEWLINE\[NEWLINE\bigg|\bigg(\frac{1}{zf(z)}\bigg)^{(n+2)}\bigg|\leq K (z\in E),NEWLINE\]NEWLINE where \(K<\frac{1-(n+2)|a_n|}{2}\) and \(1-(n+2)|a_n|>0\), then \(f(z)\) is meromorphic univalent in \(D\).