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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.

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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References:

[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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