Some properties of analytic functions relating to the Miller and Mocanu result. (English) Zbl 1217.30017

Summary: Let \(\mathcal P(\alpha)\) be the class of analytic functions \(p(z)\) in the open unit disc \(\mathbb{U}\) with \(p(0)=1\) and \(|\mathrm{arg}p(z)|<\frac{\pi}{2}\alpha\) \(z\in\mathbb{U}\) for some real \(\alpha >0\). The object of the present paper is to discuss some properties of \(p(z)\) in the class \(\mathcal P(\alpha)\). Furthermore, an example for our results is shown.


30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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[1] Duren, P.L., Univalent functions, (1983), Springer-Verlag New York, Berlin, Heidelberg, Tokyo · Zbl 0398.30010
[2] Miller, S.S.; Mocanu, P.T., Marx – strohhäcker differential subordination systems, Proc. amer. math. soc., 99, 527-533, (1987) · Zbl 0622.30020
[3] Nunokawa, M., On the order of strongly starlikeness of strongly convex functions, Proc. Japan acad., 69, 234-237, (1993) · Zbl 0793.30007
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