Summary: For a fixed analytic univalent function \(\phi\), the class of meromorphic univalent \(\alpha\)-convex functions with respect to \(\phi\) is introduced. A representation theorem for functions in this class as well as a necessary and sufficient condition for functions to belong to this class are obtained. Also we obtain a sharp growth theorem and an estimate of a certain coefficient functional for meromorphic starlike functions wtih respect to \(\phi\). Differential subordination and superordination conditions are also obtained for the subclass of meromorphic starlike functions with respect to \(\phi\).