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Subclasses of univalent functions. (English) Zbl 0531.30009
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.]
This paper is concerned with the classes $$S_ n(\alpha)=\{f:$$ f is holomorphic in the unit disk U, $$f(0)=f'(0)-1=0$$ and $$Re[D^{n+1}f(z)/D^ nf(z)]>\alpha$$ for $$z\in U\}$$, $$0\leq \alpha<1$$, where $$D^ 0f(z)=f(z),\quad D^ 1f(z)=Df(z)=zf'(z)$$ and $$D^ nf(z)=D(D^{n-1}f(z)),$$ $$n\geq 2$$. Using subordination techniques the sharp result is obtaind that $$S_{n+1}(\alpha)\subset S_ n(\delta(\alpha)),\quad 0\leq \alpha<1,$$ where $$\delta(\alpha)=(2\alpha - 1)/[2(1-2^{1-2\alpha})],\quad \alpha \neq \frac{1}{2},$$ and $$\delta(\alpha)=1/(2 \ln 2),\quad \alpha =\frac{1}{2}.$$ From a corollary it is noted that for $$0\leq \alpha<1$$, all functions in $$S_ n(\alpha)$$ 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 H. Silverman and E. M. Silvia [Rocky Mt. J. Math. 10, 469-474 (1980; Zbl 0455.30011)].
Reviewer: D.V.V.Wend

##### MSC:
 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) 30C50 Coefficient problems for univalent and multivalent functions of one complex variable 30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination