Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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REGIONS OF VARIABILITY FOR GENERALIZED α-CONVEX AND β-STARLIKE FUNCTIONS, AND THEIR EXTREME POINTS

  • Chen, Shaolin (DEPARTMENT OF MATHEMATICS HUNAN NORMAL UNIVERSITY) ;
  • Huang, Aiwu (DEPARTMENT OF MATHEMATICS HUNAN UNIVERSITY OF CHINESE MEDICINE)
  • Received : 2008.10.16
  • Published : 2010.10.31
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Abstract

Suppose that n is a positive integer. For any real number $\alpha$($\beta$ resp.) with $\alpha$ < 1 ($\beta$ > 1 resp.), let $K^{(n)}(\alpha)$ ($K^{(n)}(\beta)$ resp.) be the class of analytic functions in the unit disk $\mathbb{D}$ with f(0) = f'(0) = $\cdots$ = $f^{(n-1)}(0)$ = $f^{(n)}(0)-1\;=\;0$, Re($\frac{zf^{n+1}(z)}{f^{(n)}(z)}+1$) > $\alpha$ (Re($\frac{zf^{n+1}(z)}{f^{(n)}(z)}+1$) < $\beta$ resp.) in $\mathbb{D}$, and for any ${\lambda}\;{\in}\;\bar{\mathbb{D}}$, let $K^{(n)}({\alpha},\;{\lambda})$ $K^{(n)}({\beta},\;{\lambda})$ resp.) denote a subclass of $K^{(n)}(\alpha)$ ($K^{(n)}(\beta)$ resp.) whose elements satisfy some condition about derivatives. For any fixed $z_0\;{\in}\;\mathbb{D}$, we shall determine the two regions of variability $V^{(n)}(z_0,\;{\alpha})$, ($V^{(n)}(z_0,\;{\beta})$ resp.) and $V^{(n)}(z_0,\;{\alpha},\;{\lambda})$ ($V^{(n)}(z_0,\;{\beta},\;{\lambda})$ resp.). Also we shall determine the extreme points of the families of analytic functions which satisfy $f(\mathbb{D})\;{\subset}\;V^{(n)}(z_0,\;{\alpha})$ ($f(\mathbb{D})\;{\subset}\;V^{(n)}(z_0,\;{\beta})$ resp.) when f ranges over the classes $K^{(n)}(\alpha)$ ($K^{(n)(\beta)$ resp.) and $K^{(n)}({\alpha},\;{\lambda})$ ($K^{(n)}({\beta},\;{\lambda})$ resp.), respectively.

Keywords

Acknowledgement

Supported by : NSFs of China, NCET

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