A Landau's theorem in several complex variables

In one complex variable, it is well known that if we consider the family of all holomorphic functions on the unit disc that fix the origin and with first derivative equal to 1 at the origin, then there exists a constant ρ, independent of the functions, such that in the image of the unit disc of any of the functions of the family, there is a disc of universal radius ρ. This is the so celebrated Landau’s theorem. Many counterexamples to an analogous result in several complex variables exist. In this paper, we introduce a class of holomorphic maps for which one can get a Landau’s theorem and a Brody–Zalcman theorem in several complex variables.

In one complex variable, it is well known that if we consider the family of all holomorphic functions on the unit disc that fix the origin and with first derivative equal to 1 at the origin, then there exists a constant rho, independent of the functions, such that in the image of the unit disc of any of the functions of the family, there is a disc of universal radius rho. This is the so celebrated Landau's theorem. Many counterexamples to an analogous result in several complex variables exist. In this paper, we introduce a class of holomorphic maps for which one can get a Landau's theorem and a Brody-Zalcman theorem in several complex variables.