The first aim is to prove that the Roper-Suffridge extension operator preserves \(\varepsilon \) -starlike property on general domains given by convex functions. The second is to construct the generalized Roper-Suffridge extension operator on Reinhard domains \(\begin{aligned} \Omega _{p_{1},p_{2},\cdots ,p_{n}}=\{(z_1,\ldots ,z_{n})\in {\mathbb {C}}^{n}:\sum \limits _{j=1}^{n}|z_{j}|^{p_{j}}<1\},\quad p_{1},\cdots ,p_{n}\ge 1. \end{aligned}\) By using a refined Schwarz-Pick lemma, we prove that the operator preserves important properties, e.g., subordination property and spirallikeness. This solves a problem of Gong and Liu. Our result improves many known results from \(p_1=2\) to \(1\le p_1<\infty \) . As applications, we obtain growth and covering results associated with the extension operator on p-unit ball. Finally, by obtaining geometric and analytic properties of bounded symmetric domains, we generalize the Pfaltzgraff-Suffridge extension operator over bounded symmetric domains and prove Loewner chains and starlikeness are also preserved with a new idea. Further, we propose two conjectures for convexity property.
