Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

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

Cited in: Zbl 1240.30044 Zbl 1199.30155 Zbl 1152.30309 Zbl 1018.30009 Zbl 0863.30012 Zbl 0796.30010 Zbl 0703.30012 Zbl 0676.30015 Zbl 0623.30013 Zbl 0614.30016 Zbl 0581.30010

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]