A Bloch-Landau Theorem for slice regular functions

Abstract. The Bloch-Landau Theorem is one of the basic results in the geometric theory of holomorphic functions. It establishes that the image of the open unit disc D under a holomorphic function f (such that f (0) = 0 and f ′ (0) = 1) always contains an open disc with radius larger than a universal constant. In this paper we prove a Bloch-Landau type Theorem for slice regular functions over the skew field H of quaternions. This result is not at all a direct extension of the complex one, but heavily resents of the peculiarities of the quaternionic setting.