Two Generalizations of Bohr Radius

The purpose of this paper is twofold. First, by using the hyperbolic metric, we establish the Bohr radius for analytic functions from shifted disks containing the unit disk D into convex proper domains of the complex plane. As a consequence, we generalize the Bohr radius of Evdoridis, Ponnusamy and Rasila based on geometric idea. By introducing an alternative multidimensional Bohr radius, the second purpose is to obtain the Bohr radius of higher dimensions for Carathéodory families in the unit ball B of a complex Banach space X. Notice that when B is the unit ball of the complex Hilbert space X, we show that the constant 1/3 is the Bohr radius for normalized convex mappings of B, which generalizes the result of convex functions on D.