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
$$\begin{eqnarray}\unicode[STIX]{x1D6FA}_{N}(D)=\bigg\{(z_{1},w_{1},\ldots ,w_{k})\in \mathbb{C}\times \mathbb{C}^{n_{1}}\times \cdots \times \mathbb{C}^{n_{k}}:\Vert w_{1}\Vert _{p_{1}}^{p_{1}}+\cdots +\Vert w_{k}\Vert _{p_{k}}^{p_{k}}<\frac{1}{\unicode[STIX]{x1D706}_{D}(z_{1})}\bigg\},\end{eqnarray}$$
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
$\unicode[STIX]{x1D706}_{D}$
is the density of the hyperbolic metric on
$D$
. In this paper, we prove that
$$\begin{eqnarray}\unicode[STIX]{x1D6F7}_{N,1/p_{1},\ldots ,1/p_{k}}(f)(z_{1},w_{1},\ldots ,w_{k})=(f(z_{1}),(f^{\prime }(z_{1}))^{1/p_{1}}w_{1},\ldots ,(f^{\prime }(z_{1}))^{1/p_{k}}w_{k})\end{eqnarray}$$
is a normalized convex (resp., starlike) mapping on
$\unicode[STIX]{x1D6FA}_{N}(D)$
. If
$D$
is the unit disk, then our result reduces to Gong and Liu 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
$\unicode[STIX]{x1D6F7}_{N,1/p_{1},\ldots ,1/p_{k}}(f)$
is a spiral-like mapping of type
$\unicode[STIX]{x1D6FC}$
when
$f$
is a spiral-like function of type
$\unicode[STIX]{x1D6FC}$
on the unit disk.