In this talk, we will generalize the Roper-Suffridge extension operator which maps a biholomorhic function f on D to a biholomorphic mapping F on $$ \Omega_{n,p_{2},\cdots,p_{n}}(D)=\left\{(z_1,z_0)\in D\times{\mathbb{C}}^{n-1}: \sum\limits_{j=2}^{n}|z_{j}|^{p_{j}}<\frac{1}{\lambda_{D}(z_1)}\right\},\,\,\,p_j\geq1, $$ where $z_0=(z_2,\ldots,z_n)$ and $\lambda_{D}$ is the density of the Poincar$\acute{e}$ metric on a simply connected domain $D\subset\mathbb C$. We prove this Roper-Suffridge extension operator preserves E-starlike mapping. As a consequence, we solve a problem of Graham and Kohr in a new method. By introducing the scaling new method, the second part is to construct some new convex mappings of egg domains and obtain extemal points. By investigating the properties of the Poincar$\acute{e}$ metric, in the third part, we will consider the modified Roper-Suffridge extension operator and prove the operator preserves Loewner chain under some natrural conditions. Finally, we propose some problems. This work is jointed with Prof. Taishun Liu and Prof. Xiaomin Tang.