## 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.)

### Keywords:

strongly starlike of order $$\beta$$
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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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