In this paper, we first give the distortion and growth theorems for convex functions \(f(\zeta )\) on the unit disc \(\mathbb {D}\) in \(\mathbb {C}\) which have a g-parametric and satisfy that \(\zeta =0\) is a zero of order \(k+1\) of \(f(\zeta )-\zeta \) , where \(g(\zeta )=\frac{1+E\zeta }{1+F\zeta }, -1\leqslant F<E\leqslant 1,\,\,\zeta \in \mathbb {D}\) . Next, we extend these corresponding theorems to the case of some subclasses of normalized quasi-convex mappings of type \(\mathbb {B}\) on the unit ball of a complex Banach space (resp. normalized quasi-convex mappings of type \(\mathbb {A}\) on the unit polydisc and unit ball with the arbitrary norm in \(\mathbb {C}^n\) ). Our theorems further partially solve the Gong’s conjecture in several complex variables. Several particular cases will be also discussed.
