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On the order of strongly starlikeness of strongly convex functions. (English) Zbl 0793.30007
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.

30C45Special classes of univalent and multivalent functions
Full Text: DOI
[1] P. T. Mocanu: Alpha-convex integral operator and strongly starlike functions. Studia Univ. Babes-Bolyai Mathematica, 34, 2, 18-24 (1989). · Zbl 0900.30012
[2] M. Nunokawa: On properties of non-Caratheodory functions. Proc. Japan Acad., 68A, 152-153 (1992). · Zbl 0773.30020 · doi:10.3792/pjaa.68.152