Distortion Theorems for Almost Convex Mappings of Order \alpha in Several Complex Variables

In this paper, first, a sufficient condition for almost convex mappings of order \(\alpha \) defined on the unit ball of complex Hilbert spaces and another sufficient condition for almost quasi-convex mappings of order \(\alpha \) defined on the unit ball of complex Banach spaces are given. Second, the distortion theorem of the Frechet derivative for almost convex mappings of order \(\alpha \) on the unit ball of complex Banach spaces, the homogeneous ball of complex Banach spaces, and the unit ball of complex Hilbert spaces are established respectively. Finally, the distortion theorem of the Jacobi determinant for almost convex mappings of order \(\alpha \) on the Euclidean unit ball in \(\mathbb {C}^n\) is obtained. Our results generalize many known results.