Fekete-szegö results for certain class of meromorphic functions using \(q-\)derivative operator

Abstract

In the present paper, we introduce the subclasses \(\sum _{b}^{*}\left( q,\phi \right) \) and \(\sum _{b}^{*}\left( \alpha ,q,\phi \right) \) of meromorphic functions \(f\left( z\right) \) satisfying \(1+\frac{1}{b}\left[ -\frac{qzD_{q}^{*}f(z)}{f(z)}-1\right] \prec \phi (z)\) and \(1+\frac{1}{b}\left[ \frac{-\left( 1-\frac{\alpha }{q}\right) qzD_{q}^{*}f\left( z\right) +\alpha qzD_{q}^{*}\left[ zD_{q}^{*}f\left( z\right) \right] }{\left( 1-\frac{\alpha }{q}\right) f\left( z\right) -\alpha zD_{q}^{*}f\left( z\right) }-1\right] \prec \phi (z)\ (b\in \mathbb {C} ^{*}=\mathbb {C}\backslash \left\{ 0\right\} ,\ \) \(\alpha \in \mathbb {C}\backslash (0,1],\ \operatorname {Re}(\alpha )\ge 0,\ 0<q<1)\), respectively. Sharp bounds for the Fekete-Szegö functional \(\left| a_{1}-\mu a_{0}^{2}\right| \) are obtained.