Abstract
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.
Citation
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. https://doi.org/10.36045/bbms/1274896206
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