## A Landau’s theorem in several complex variables.(English)Zbl 1366.32008

One can show, by example, that the Landau theorem and the related result of Brody-Zalcman is not true in several variables.
In this paper, the author introduces a class of holomorphic maps for which one can get a Landau theorem and a Brody-Zalcman theorem in several complex variables.
Denote by $$B^k=\big\{z\in \mathbb C^k : \sum_{j=1}^k|z_j|^2<1\big\}$$ the unit ball in $$\mathbb C^k$$, $$k>1$$. Let $$\Phi: B^k \rightarrow \mathbb C^k$$ be a holomorphic map. Let $$\Phi^\prime(a)$$ denote the Jacobian matrix of $$\Phi$$ computed in the point $$a\in B^k.$$
The author proves the following theorems.
Theorem 2.1. (Brody-Zalcman-type theorem) Let $$C$$ be a positive constant. Consider a family of holomorphic maps $$\Phi$$ satisfying $||\Phi^\prime(a)|| \cdot ||\Phi^\prime(a)^{-1}|| \leq C,\quad \forall a \in B^k.$ If the family is not normal, then, after reparametrization, we can extract a subsequence converging to a non-degenerate holomorphic map $$\Psi : \mathbb C^k \rightarrow \mathbb C^k$$.
Assume that $$\Phi: B^k \rightarrow\mathbb C^k$$, $$\Phi(0) = 0$$, and $$\Phi^\prime(0) = \mathrm{Id}$$.
Theorem 3.1. (Landau-type theorem) Let $$C$$ be a positive constant such that $||\Phi^\prime(z)|| \cdot ||\Phi^\prime(z)^{-1}|| \leq C,\quad \forall z \in B^k.$ Then there exists $$\rho>0$$, depending only on $$C$$, such that $$\Phi(B^k)$$ contains a ball of radius $$\rho>0$$. There exists also a domain $$U$$ such that $$\Phi(U) = B(a,\rho)$$.

### MSC:

 32H99 Holomorphic mappings and correspondences 32A18 Bloch functions, normal functions of several complex variables
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### References:

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