Summary: Suppose that \(D\subset \mathbb{C}\) is a simply connected subdomain containing the origin and \(f(z_1)\) is a normalized convex (resp., starlike) function on \(D\). Let \[ \Omega_N(D)=\bigg\{(z_1,w_1,\ldots ,w_k)\in \mathbb{C}\times \mathbb{C}^{n_1}\times \cdots \times \mathbb{C}^{n_k}:\Vert w_1\Vert _{p_1}^{p_1}+\cdots +\Vert w_k\Vert _{p_k}^{p_k}<\frac{1}{\lambda_D(z_1)}\bigg\}, \] where \(p_j\geqslant 1\), \(N=1+n_1+\cdots +n_k\), \(w_1\in \mathbb{C}^{n_1},\ldots ,w_k\in \mathbb{C}^{n_k}\) and \(\lambda_D\) is the density of the hyperbolic metric on \(D\). In this paper, we prove that \[ \Phi_{N,1/p_1,\ldots ,1/p_k}(f)(z_1,w_1,\ldots ,w_k)=(f(z_1),(f^{\prime }(z_1))^{1/p_1}w_1,\ldots ,(f^{\prime}(z_1))^{1/p_k}w_k) \] is a normalized convex (resp., starlike) mapping on \(\Omega_N(D)\). If \(D\) is the unit disk, then our result reduces to [S. Gong and T. Liu, J. Math. Anal. Appl. 284, No. 2, 425–434 (2003; Zbl 1031.32002)] via a new method. Moreover, we give a new operator for convex mapping construction on an unbounded domain in \(\mathbb{C}^2\). Using a geometric approach, we prove that \(\Phi_{N,1/p_1,\ldots ,1/p_k}(f)\) is a spiral-like mapping of type \(\alpha\) when \(f\) is a spiral-like function of type \(\alpha\) on the unit disk.