Sufficient conditions for functions to be meromorphic starlike of reciprocal order \(\alpha\)

A meromorphic function \(f(z)=z^{-1}+a_0+a_1 z+...\) $f(z)=z^{-1}+a_0+a_{1}z+...$ is starlike of reciprocal order \(\alpha\) $\alpha$ if \(-{{\,\textrm{Re}\,}}\left( f(z)/(zf'(z))\right) > \alpha\) $-\text{Re}\left(f,(z),/,(,z,f^{\prime },(z),)\right)>\alpha$ for all z in the unit disc. Using the theory of first order differential subordination an admissible condition is obtained so that \(\Psi (\mathcal {Q}_{ST}(z),\mathcal {Q}_{CV}(z);z)\in \Omega\) $\mathrm{\Psi }({\mathcal{Q}}_{\mathit{ST}}(z),{\mathcal{Q}}_{\mathit{CV}}(z);z)\in \mathrm{\Omega }$ where \(\mathcal {Q}_{ST}(z)=-(zf'(z)/f(z))\) ${\mathcal{Q}}_{\mathit{ST}}(z)=-(z{f}^{\prime }(z)/f(z))$ and \(\mathcal {Q}_{CV}(z)=-(1+(zf''(z))/f'(z))\) ${\mathcal{Q}}_{\mathit{CV}}(z)=-(1+(z{f}^{\prime \prime }(z))/f^{\prime }(z))$ implies that f is starlike of reciprocal order \(\alpha\) $\alpha$ . Applying this result, various sufficient conditions are obtained for functions belonging to the class of all meromorphic starlike functions of reciprocal order \(\alpha\) $\alpha$ , for \(0<\alpha < 1\) $0<\alpha <1$ . Some sufficient conditions for functions f are obtained so that the image of \(f(z)/(z f'(z))\) $f(z)/(z{f}^{\prime }(z))$ lies inside the disc centered at \(-1\) $-1$ with radius \((1 - \alpha )\) $(1-\alpha )$ .