Loewner chain associated with the generalized Graham-Kohr extension operator

Abstract 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 $$\mathbb {B}$$ .