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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 \[ |\arg(1+ zf''(z)/f'(z)|< \pi\gamma/2\quad\text{for all }z\in E, \] then \(|\arg zf'(z)/f(z)|<\pi\beta/2\) 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.

MSC:

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

[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
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