Regions of variability for generalized $\alpha$-convex and $\beta$-starlike functions, and their extreme points
Commun. Korean Math. Soc. 2010 Vol. 25, No. 4, 557-569
https://doi.org/10.4134/CKMS.2010.25.4.557
Printed December 1, 2010
Shaolin Chen and Aiwu Huang
Hunan Normal University, Hunan University of Chinese Medicine
Abstract : Suppose that $n$ is a positive integer. For any real number $\alpha$ ($\beta$ resp.) with $\alpha1$ resp.), let $K^{\langle n \rangle}(\alpha)$ ($K^{\langle n \rangle}(\beta)$ 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 $\mathcal D$, and for any $\lambda$,
Keywords : Schwarz lemma, analytic function, univalent function, starlike function, generalized $\alpha$-convex domain, $\beta$-starlike function, region of variability, extreme point
MSC numbers : Primary 30C65, 30C45; Secondary 30C20
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