Application of duality techniques to starlikeness of weighted integral transforms

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.