Application of duality techniques to starlikeness of weighted integral transforms
Ebadian, A. ; Aghalary, R. ; Shams, S.
Bull. Belg. Math. Soc. Simon Stevin, Tome 17 (2010) no. 1, p. 275-285 / Harvested from Project Euclid
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.
Publié le : 2010-04-15
Classification:  Convolution,  Univalent functions,  Hypergeometric function,  Duality technique,  Starlike functions,  30C45,  30C80
@article{1274896206,
     author = {Ebadian, A. and Aghalary, R. and Shams, S.},
     title = {Application of duality techniques to starlikeness of weighted integral transforms},
     journal = {Bull. Belg. Math. Soc. Simon Stevin},
     volume = {17},
     number = {1},
     year = {2010},
     pages = { 275-285},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1274896206}
}
Ebadian, A.; Aghalary, R.; Shams, S. Application of duality techniques to starlikeness of weighted integral transforms. Bull. Belg. Math. Soc. Simon Stevin, Tome 17 (2010) no. 1, pp.  275-285. http://gdmltest.u-ga.fr/item/1274896206/