Loewner chain associated with the generalized Graham-Kohr extension operator

Abstract Let f be a locally univalent function defined on the unit disc U, and let ₙ 0, 1 and ₙ 0, 12. We consider the family of operators extending f to a holomorphic mapping from the unit ball B in Cⁿ to Cⁿ given by: ₍, 䂸, 䂸 (f) (z) = (f (z₁), (f (z₁) z₁) ^ ₙ (f^ (z₁) ) ^ ₙz^) where z= (z₁, z^) Cⁿ and z^ = (z₂, , zₙ), n 2. When ₙ=12 and ₙ=0, this operator coincides with the classical Roper-Suffridge extension operator. We first prove that the operator ₍, 䂸, 䂸 maps the family of spirallike functions of type (denoted by S_) into the class of mappings that have parametric representation on Bⁿ (denoted by S⁰ (Bⁿ) ). In the second part, we show that if f is a normalized univalent Bloch function on the unit disc U, then ₍, , (f) is a Bloch mapping on the unit ball B.