Let f be a locally univalent function defined on the unit disc \(\mathbb {U}\) , and let \(\gamma _n \in [0,1]\) and \(\omega _n \in \left[ 0, \frac{1}{2}\right] \) . We consider the family of operators extending f to a holomorphic mapping from the unit ball \(\mathbb {B}\) in \(\mathbb {C}^n\) to \(\mathbb {C}^n\) given by: \( \Theta _{n, \gamma _n, \omega _n}(f)(z)=\left( f\left( z_1\right) ,\left( \frac{f\left( z_1\right) }{z_1}\right) ^{\gamma _n}\left( f^{\prime }\left( z_1\right) \right) ^{\omega _n}z^{\prime }\right) \) where \(z=\left( z_1, z^{\prime }\right) \in \mathbb {C}^n\) and \(z^{\prime }=\left( z_2, \ldots , z_n\right) \) , \(n\ge 2\) . When \(\omega _n=\frac{1}{2}\) and \(\gamma _n=0\) , this operator coincides with the classical Roper-Suffridge extension operator. We first prove that the operator \(\Theta _{n,\gamma _n,\omega _n}\) maps the family of spirallike functions of type \(\beta \) (denoted by \(\hat{S}_\beta \) ) into the class of mappings that have parametric representation on \(\mathbb {B}^n\) (denoted by \(S^0 (\mathbb {B}^n)\) ). In the second part, we show that if f is a normalized univalent Bloch function on the unit disc \(\mathbb {U}\) , then \(\Theta _{n,\gamma ,\omega }(f)\) is a Bloch mapping on the unit ball \(\mathbb {B}\) .
