REGIONS OF VARIABILITY FOR GENERALIZED α-CONVEX AND β-STARLIKE FUNCTIONS, AND THEIR EXTREME POINTS

Title & Authors
REGIONS OF VARIABILITY FOR GENERALIZED α-CONVEX AND β-STARLIKE FUNCTIONS, AND THEIR EXTREME POINTS
Chen, Shaolin; Huang, Aiwu;

Abstract
Suppose that n is a positive integer. For any real number $\small{\alpha}$($\small{\beta}$ resp.) with $\small{\alpha}$ < 1 ($\small{\beta}$ > 1 resp.), let $\small{K^{(n)}(\alpha)}$ ($\small{K^{(n)}(\beta)}$ resp.) be the class of analytic functions in the unit disk $\small{\mathbb{D}}$ with f(0) = f'(0) = $\small{\cdots}$ = $\small{f^{(n-1)}(0)}$ = $\small{f^{(n)}(0)-1\;=\;0}$, Re($\small{\frac{zf^{n+1}(z)}{f^{(n)}(z)}+1}$) > $\small{\alpha}$ (Re($\small{\frac{zf^{n+1}(z)}{f^{(n)}(z)}+1}$) < $\small{\beta}$ resp.) in $\small{\mathbb{D}}$, and for any $\small{{\lambda}\;{\in}\;\bar{\mathbb{D}}}$, let $\small{K^{(n)}({\alpha},\;{\lambda})}$ $\small{K^{(n)}({\beta},\;{\lambda})}$ resp.) denote a subclass of $\small{K^{(n)}(\alpha)}$ ($\small{K^{(n)}(\beta)}$ resp.) whose elements satisfy some condition about derivatives. For any fixed $\small{z}$$\small{_0\;{\in}\;\mathbb{D}}$, we shall determine the two regions of variability $\small{V^{(n)}(z_0,\;{\alpha})}$, ($\small{V^{(n)}(z_0,\;{\beta})}$ resp.) and $\small{V^{(n)}(z_0,\;{\alpha},\;{\lambda})}$ ($\small{V^{(n)}(z_0,\;{\beta},\;{\lambda})}$ resp.). Also we shall determine the extreme points of the families of analytic functions which satisfy $\small{f(\mathbb{D})\;{\subset}\;V^{(n)}(z_0,\;{\alpha})}$ ($\small{f(\mathbb{D})\;{\subset}\;V^{(n)}(z_0,\;{\beta})}$ resp.) when f ranges over the classes $\small{K^{(n)}(\alpha)}$ ($\small{K^{(n)(\beta)}$ resp.) and $\small{K^{(n)}({\alpha},\;{\lambda})}$ ($\small{K^{(n)}({\beta},\;{\lambda})}$ resp.), respectively.
Keywords
Schwarz lemma;analytic function;univalent function;starlike function;generalized $\small{\alpha}$-convex domain;$\small{\beta}$-starlike function;region of variability;extreme point;
Language
English
Cited by
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