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april 2010 Application of duality techniques to starlikeness of weighted integral transforms
R. Aghalary, A. Ebadian, S. Shams
Bull. Belg. Math. Soc. Simon Stevin 17(2): 275-285 (april 2010). DOI: 10.36045/bbms/1274896206


Let $\mathcal{A}$ be the class of normalized analytic functions in the unit disc and let $P_{\gamma}(\alpha, \beta)$ be the class of all functions $f \in \mathcal{A}$ satisfying the condition \[ \exists \ \eta \in \mathbb{R}, \quad \Re \left \{ e^{i \eta}\left[(1-\gamma)\left(\frac{f(z)}{z}\right)^{\alpha} + \gamma \frac{zf'(z)}{f(z)}\left(\frac{f(z)}{z}\right)^{\alpha} - \beta \right] \right \} 0 .\] We consider the integral transform \[ V_{\lambda, \alpha}(f)(z)=\left\{\int_{0}^{1}\lambda(t) \left(\frac{f(tz)}{t} \right)^{\alpha} dt\right\}^{\frac{1}{\alpha}},\] where $\lambda(t)$ is a real-valued nonnegative weight function normalized by\linebreak $\int_{0}^{1}\lambda(t) dt=1$. In this paper we find conditions on the parameters $\alpha, \beta, \gamma, \mu $ such that $V_{\lambda, \alpha}(f)$ maps $P_{\gamma}(\alpha, \beta)$ into the class of starlike functions of order $\mu$. We also provide a number of applications for various choices of $\lambda(t)$. Our results generalize known results on this topic.


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R. Aghalary. A. Ebadian. S. Shams. "Application of duality techniques to starlikeness of weighted integral transforms." Bull. Belg. Math. Soc. Simon Stevin 17 (2) 275 - 285, april 2010.


Published: april 2010
First available in Project Euclid: 26 May 2010

zbMATH: 1194.30009
MathSciNet: MR2663473
Digital Object Identifier: 10.36045/bbms/1274896206

Primary: 30C45
Secondary: 30C80

Keywords: convolution , Duality technique , hypergeometric function , Starlike functions , univalent functions

Rights: Copyright © 2010 The Belgian Mathematical Society


Vol.17 • No. 2 • april 2010
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