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Zbl 0793.30007
Nunokawa, Mamoru
On the order of strongly starlikeness of strongly convex functions. (English)
[J] Proc. Japan Acad., Ser. A 69, No.7, 234-237 (1993). ISSN 0386-2194

Let $A$ denote the set of functions $f(z)$ analytic in the unit disc $E$ with $f(0)=0$ and $f'(0)=1$. P. T. Mocanu has proved that if $$\vert\arg(1+ zf''(z)/f'(z)\vert< \pi\gamma/2\quad\text{for all }z\in E,$$ then $\vert\arg zf'(z)/f(z)\vert<\pi\beta/2$ there, where $\gamma$ and $\beta$ are between 0 and 1 and are related by a somewhat complicated functional relation. That is, strongly convex of order $\gamma$ implies strongly starlike of order $\beta$. This paper proves the same result with a more complicated functional relationship between $\gamma$ and $\beta$. Unfortunately, numerical calculations appear to indicate that the two sets of relationships give exactly the same $(\gamma,\beta)$ pairs to at least ten decimal places. The proof here seems to be different from that of Mocanu.
[J.A.Hummel (College Park)]

MSC 2000:: *30C45 Special classes of univalent and multivalent functions

Keywords: strongly starlike of order $\beta$

Cited in: Zbl 1225.30015 Zbl 1224.30066 Zbl 1195.30030 Zbl 1194.30013 Zbl 1086.30018 Zbl 0880.30013

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