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Classes of meromorphic \(\alpha\)-convex functions. (English) Zbl 1213.30006

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 \(\phi\).

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination
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