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Zbl 0531.30009
Sălăgean, Grigore Stefan
Subclasses of univalent functions.
(English)
[A] Complex analysis - Proc. 5th Rom.-Finn. Semin., Bucharest 1981, Part 1, Lect. Notes Math. 1013, 362-372 (1983).

[For the entire collection see Zbl 0516.00016.] \par This paper is concerned with the classes $S\sb n(\alpha)=\{f:$ f is holomorphic in the unit disk U, $f(0)=f'(0)-1=0$ and $Re[D\sp{n+1}f(z)/D\sp nf(z)]>\alpha$ for $z\in U\}$, $0\le \alpha<1$, where $D\sp 0f(z)=f(z),\quad D\sp 1f(z)=Df(z)=zf'(z)$ and $D\sp nf(z)=D(D\sp{n-1}f(z)),$ $n\ge 2$. Using subordination techniques the sharp result is obtaind that $S\sb{n+1}(\alpha)\subset S\sb n(\delta(\alpha)),\quad 0\le \alpha<1,$ where $\delta(\alpha)=(2\alpha - 1)/[2(1-2\sp{1-2\alpha})],\quad \alpha \ne \frac{1}{2},$ and $\delta(\alpha)=1/(2 \ln 2),\quad \alpha =\frac{1}{2}.$ From a corollary it is noted that for $0\le \alpha<1$, all functions in $S\sb n(\alpha)$ are starlike for n a nonnegative integer and convex for n a positive integer. The author also obtains coefficients bounds that generalize a result of {\it H. Silverman} and {\it E. M. Silvia} [Rocky Mt. J. Math. 10, 469-474 (1980; Zbl 0455.30011)].
[D.V.V.Wend]
MSC 2000:
*30C45 Special classes of univalent and multivalent functions
30C50 Coefficient problems for univalent and multivalent functions
30C80 Maximum principle, etc. (one complex variable)

Keywords: subclass of univalent functions; starlike functions; subordination

Citations: Zbl 0516.00016; Zbl 0455.30011

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