Loewner chain associated with the generalized Graham-Kohr extension operator

Let f be a locally univalent function defined on the unit disc U , and let γ ₙ ∈ [0,1] and ω ₙ ∈ [ 0, 1/2] . We consider the family of operators extending f to a holomorphic mapping from the unit ball B in Cⁿ to Cⁿ given by: $$ Θ n, γ _n, ω _n(f)(z)=( f( z_1) ,( f( z_1) /z_1) γ _n( f( z_1) ) ω _nz) $$ where $$z=( z_1, z) ∈ C^n$$ and $$z=( z_2, … , z_n) $$ , $$n≥ 2$$ . When $$ω _n=1/2$$ and $$γ _n=0$$ , this operator coincides with the classical Roper-Suffridge extension operator. We first prove that the operator $$Θ n,γ _n,ω _n$$ maps the family of spirallike functions of type $$β $$ (denoted by $$Ŝ_β $$ ) into the class of mappings that have parametric representation on $$B^n$$ (denoted by $$S^0 (B^n)$$ ). In the second part, we show that if f is a normalized univalent Bloch function on the unit disc $$U$$ , then $$Θ n,γ ,ω(f)$$ is a Bloch mapping on the unit ball $$B$$ .