Language:   Search:   Contact
Zentralblatt MATH has released its new interface!
For an improved author identification, see the new author database of ZBMATH.

Query:
Fill in the form and click »Search«...
Format:
Display: entries per page entries
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 $\beta$

Highlights
Master Server