Let \(\mathbb{C}^{n}\) be the space of \(n\)-complex variables \(z=(z_{1},\dots,z_{n})\) with the usual inner product \(\langle z,w\rangle= \sum_{j=1}^{n}z_{j} \overline{w}_{j}\), \(w=(w_{1},\dots ,w_{n})\in\mathbb{C}^{n}\). Consider the norm \(\| z\| _{p}= (\sum_{j=1}^{n} | z_{j}| ^{p})^{1/p}\), \(z\in \mathbb{C}^{n}\), \(p>1\) and let \(B_{p}^{n}= \{z\in\mathbb{C}^{n}:\| z\| _{p}<1\}\) be the Reinhardt domain. Let \(H(B_{p}^{n})\) be the class of holomorphic mappings from \(B_{p}^{n}\) into \(\mathbb{C}^{n}\). A mapping \(f\in H(B_{p}^{n})\) is said to be local biholomorphic in \(B_{p}^{n}\) if \(f\) has a local inverse at each point \(z\in B_{p}^{n}\). A biholomorphic convex mapping on \(B_{p}^{n}\) is a biholomorphic mapping on \(B_{p}^{n}\) such that \(f(B_{p}^{n})\) is a convex domain in \(\mathbb{C}^{n}\). Let \(K(B_{p}^{n})\) denote the class of all biholomorphic convex mappings \(f\) on \(B_{p}^{n}\) with \(f(0)=0\), \(Df(0)=I\).
In the paper the authors prove some sufficient conditions for a biholomorphic mapping to be in the class \(K(B_{p}^{n})\). Some concrete biholomorphic convex mappings on \(B_{p}^{n}\) are constructed.