1 Introduction

An important tool in geometric function theory in one complex variable is the so-called Landau’s theorem.

It says: if we consider the family \({\mathcal {F}}\), of holomorphic functions in the unit disc \(\Delta \) of \({\mathbb {C}}\) such that \(f(0)=0, f'(0)=1\), then there is a constant \(\rho >0\), independent of f, such that \(f(\Delta )\) contains a disc of radius \(\rho \) [8, 16–18].

By the Landau’s number l(f) of f is meant the supremum of the set of positive numbers r such that \(f(\Delta )\) contains a disc of radius r. By the Landau’s constant \(\rho \), we meant \(\inf \nolimits _{f \in {\mathcal {F}}} l(f).\)

An easy consequence of such result is the Picard theorem: a non-constant entire function \(f{:}\,{\mathbb {C}} \rightarrow {\mathbb {C}}\) can omit at most one point.

A related result is the compactness result of Brody–Zalcman: if \({\mathcal {H}}\) is a non-relative compact family of holomorphic maps in the unit disc, then after reparametrization we can get an extracted subsequence of functions in \({\mathcal {H}}\) converging to a non-constant entire map [6, 25, 26].

It is well known that such principles break down in several complex variables in a spectacular way.

There are holomorphic maps \(F{:}\,{\mathbb {C}}^2 \rightarrow {\mathbb {C}}^2\) with Jacobian equal to 1,  and whose image omits a non-empty open set. These maps play an important role in holomorphic dynamics, see [22].

To be more explicit, let \(h(z,w)=(z^2 + aw, z),\) with \(|a|<1,\) be a so-called hénon map of \({\mathbb {C}}^2\): it fixes (0, 0) and the attraction domain \(\varOmega \) of (0, 0) is a Fatou–Bieberbach domain isomorphic to \({\mathbb {C}}^2.\) Denote with \(\varPsi _h{:}\,{\mathbb {C}}^2 \rightarrow \varOmega \) such isomorphism such that \(\varPsi ^{'}_h (0,0)=\mathrm{Id}.\) Let \(\overline{h}\) be the meromorphic extension of h to \({\mathbb {P}}^2.\) If (z, w) are the coordinates of \({\mathbb {C}}^2\) and [z : w : t] are the coordinates of \({\mathbb {P}}^2,\) then the line at infinity has equation \(\{ t=0 \}.\) We denote, respectively, with \(I^+\) and \(I^-\) the indeterminacy locus of \(\overline{h}\) and \(\overline{h^{-1}}\); they are two isolated points in \(\{ t=0 \}.\) It is well known, see [22], that the hénon map h has \(I^{-}=[1{:}0{:}0]\), and it is an attracting fixed point at the line at infinity. Then, \(\varOmega \) omits the basin of attraction of \(I^-,\) which is an open set. Hence, it cannot be that each \(\varPsi _{h,n} (z,w):=\frac{1}{n} \varPsi _h (nz,nw)\) is such that the image of B(0, 1) contains a ball B(0, r),  because if this is the case, then \(\varPsi _h (nB(0,1)) \supset n B(0,r)\) and finally \(\varPsi _h ({\mathbb {C}}^2) \supset {\mathbb {C}}^2,\) which is not true because \(\varPsi _h\) omits an open set.

There are also some elder counterexamples, one by L.A. Harris, published in 1977 [15] that we recall for the sake of completeness: given \(\delta >0,\) choose n so that \(n\delta ^2>2\) and denote \(g(z,w)=(z+nw^2,w).\) Suppose that the image of the open polydisc by g contains a ball of radius \(\delta \) and let \((\alpha _0,\beta _0)\) be its center. Then, given \(|\zeta | < \delta \), there exist points \((z_0,w_0)\) and \((z_1,w_1)\) in the polydisc such that \(g(z_0,w_0)=(\alpha _0,\beta _0)\) and \(g(z_1,w_1)=(\alpha _0,\beta _0 + \zeta ).\) Hence, \(w_1 - w_0 = \zeta \), \(w_1 + w_0= 2\beta _0 + \zeta \), and \(n(w_1^2 - w_0^2)=z_0 - z_1\), so \(n|\zeta | |2\beta _0 + \zeta | \le 2,\) for all \(|\zeta | < \delta .\) Thus, \(n\delta ^2 \le 2,\) the desired contradiction.

In the same spirit, a second counterexample was given by Duren and Rudin [9]: if \(\delta >0,\) then the map \(f(z,w)=\big (z, w + \big (\frac{z}{\delta }\big )^2\big )\) is in the class of all biholomorphic maps from the unit polydisc into \({\mathbb {C}}^2\) which fix the origin and whose Jacobian matrix is the identity at the origin, but the image under f of the polydisc contains no closed ball of radius \(\delta .\)

Indeed, for no \((u,v) \in {\mathbb {C}}^2\), the image by f of the polydisc \(\Delta ^2\) contains the circle:

$$\begin{aligned} C=\left\{ (u+\delta e^{i\theta }, v){:}\,-\pi \le \theta \le \pi \right\} . \end{aligned}$$

To see this, fix \((u,v)\in {\mathbb {C}}^2.\) If \((u +\lambda ,v) \in f(\Delta ^2)\) then, by definition of f,  we have that:

$$\begin{aligned} |v-\delta ^{-2}(u+\lambda )^2|<1. \end{aligned}$$

Therefore, if all points of C were in \(f(\Delta ^2),\) the inequality:

$$\begin{aligned} |(\delta ^2v-u^2)-2u\delta e^{i\theta } - \delta ^2 e^{2i\theta }| <\delta ^2 \end{aligned}$$

would hold for all \(\theta .\) Parseval’s equality shows that this is impossible.

Even if several weak versions of the Landau’s theorem in several complex variables have already been given, see, for instance, [7, 10, 12, 13, 19, 23, 24]; nevertheless, the author believes that this paper can add something to the already existing literature: indeed, the purpose of this note is to introduce a class of holomorphic maps for which one can get a Landau’s theorem and a Brody–Zalcman theorem in more than one variable and to underline the connection among the two.

Recently, the Landau’s theorem has also bring the attention of people working over the quaternionic variable, see, for instance, [1–5].

2 A theorem of Brody–Zalcman type

Let \(\varPhi {:}\,{\mathbb {B}}^k \rightarrow {\mathbb {C}}^k\) where \({\mathbb {B}}^k\) is the unit ball of \({\mathbb {C}}^k,\) \(k>1.\) Let \(\varPhi '(a)\) denote the Jacobian matrix of \(\varPhi \) computed in the point a.

Theorem 2.1

(Brody–Zalcman-type Theorem) Let C be a positive constant. Consider a family of such holomorphic maps \(\varPhi \) satisfying

$$\begin{aligned} ||\varPhi ^{'} (a) ||\cdot ||\varPhi ^{'} (a)^{-1} || \le C, \,\,\,\,\, \forall \,\, a \in {\mathbb {B}}^k. \end{aligned}$$
(1)

If the family is not normal, then, after reparametrization, we can extract a subsequence converging to a non-degenerate holomorphic map \(\varPsi {:}\,{\mathbb {C}}^k \rightarrow {\mathbb {C}}^k.\)

Proof

Let \(\varPhi _n\) be an arbitrary sequence of maps of the family. We can assume that the maps \(\varPhi _n\) are defined in a neighborhood of \(\overline{{\mathbb {B}}^k}.\)

Define

$$\begin{aligned} \lambda _n := \sup \limits _{|z|<1} (1-|z|) ||\varPhi _n^{'} (z)||. \end{aligned}$$

We can also assume that \(\lambda _n \rightarrow + \infty ,\) because if not the family is normal [21]. Let \(a_n\) be such that \((1-|a_n|)(||\varPhi ^{'}_n (a_n)||)=\lambda _n.\)

Define

$$\begin{aligned} B_n := [(\varPhi _n^{'})(a_n)]^{-1} \end{aligned}$$

and

$$\begin{aligned} \varPsi _n(z) := \varPhi _n (a_n + B_n z). \end{aligned}$$

We are going to show that \(\varPsi '_n (z)\) is well defined in \(|z| \le \frac{\lambda _n}{2C}.\)

Indeed, \(\varPsi _n^{'} (z) =\varPhi _n^{'} (a_n + B_n z) \circ B_n\) with \(\varPsi _n^{'} (0)=\mathrm{Id}\) and since

$$\begin{aligned} (1-|a_n + B_n z|)||\varPhi _n^{'} (a_n + B_n z)|| \le \lambda _n \end{aligned}$$

we have:

$$\begin{aligned} ||\varPsi _n^{'} (z)|| \le ||\varPhi _n^{'} (a_n + B_n z)|| \cdot ||B_n|| \le \frac{\lambda _n ||B_n||}{1-|a_n|-|B_nz|}. \end{aligned}$$
(2)

If \(|z| \le \frac{\lambda _n}{2C},\) then by (1):

$$\begin{aligned} |B_n z| \le \frac{C|z|}{||\varPhi ^{'}(a_n)||} \le \frac{(1-|a_n|)}{\lambda _n} C \frac{\lambda _n}{2C}= \frac{1-|a_n|}{2}. \end{aligned}$$
(3)

So

$$\begin{aligned} ||\varPsi _n^{'} (z)|| \le \frac{2 \lambda _n}{1-|a_n|} ||B_n|| \le \frac{2\lambda _n}{1-|a_n|} \frac{C}{||\varPhi _n^{'} (a_n)||}\le 2C \end{aligned}$$

So the family \(\varPsi _n\) is locally normal in \(|z| \le \frac{\lambda _n}{C} \rightarrow \infty .\)

We get that \(\varPsi _n\) tends to a holomorphic map \({\mathbb {C}}^k \rightarrow {\mathbb {C}}^k.\) Moreover, \(\varPsi _n^{'} (0)=\mathrm{Id}\) so \(\varPsi ^{'} (0)= \mathrm{Id}.\) Hence, \(\varPsi \) is non-degenerate. \(\square \)

Remark 2.2

If \(k=1,\) then (1) is automatically satisfied.

Remark 2.3

Condition (1) implies that all the eigenvalues are comparable, i.e., \(|\lambda _{\text {max}}(\varPhi ^{'} (a))| \le C |\lambda _{\text {min}} (\varPhi ^{'} (a))|\) for all \(a \in {\mathbb {B}}^k,\) where \(|\lambda _{\text {min}} (\varPhi ^{'} (a))|\) and \(|\lambda _{\text {max}} (\varPhi ^{'} (a))|\) are, respectively, the minimal and the maximal modulus of the eigenvalues of the Jacobian matrix \(\varPhi ^{'} (a).\)

Remark 2.4

Furthermore, it is enough to assume (1) out of an analytic set, so the maps \(\varPhi \) do not need to be locally invertible. Indeed, if we suppose that (1) holds out of an analytic set \(A_{\varPhi },\) we can choose \(a_n \notin A_{\varPhi _n}.\)

Remark 2.5

The condition (1) can be refined in

$$\begin{aligned} \sup \limits _{|z| \le \frac{1-|a|}{2}} ||\varPhi ^{'} (a+z) \cdot \varPhi ^{'} (a)^{-1}|| \le C \end{aligned}$$

This condition is less strong of (1) and implies that \(||\varPsi _n^{'}||\) is bounded for \(|z| \le d_n\) with \(d_n \rightarrow + \infty .\)

Remark 2.6

The Brody–Zalcman renormalization Theorem 2.1 works also for families of maps from \({\mathbb {B}}^k\) to a compact Hermitian manifold M of dimension k [11].

We also point out that, in [20], several sufficient conditions for a family of quasiregular mappings to be normal were previously given.

3 Landau’s theorem

Assume that \(\varPhi :{\mathbb {B}}^k \rightarrow {\mathbb {C}}^k,\) \(\varPhi (0)=0,\) and \(\varPhi ^{'} (0)= \mathrm{\mathrm{Id}}.\)

Theorem 3.1

(Landau’s theorem) Let C be a positive constant. Assume

$$\begin{aligned} ||\varPhi ^{'} (z)|| \cdot ||\varPhi ^{'} (z)^{-1}|| \le C, \,\, \forall \,\, z \in {\mathbb {B}}^k, \end{aligned}$$
(4)

then there exists \(\rho >0,\) depending only on C,  such that \(\varPhi ({\mathbb {B}}^k)\) contains a ball of radius \(\rho >0.\) There exists also a domain U such that \(\varPhi (U)=B(a, \rho ).\)

Proof

Suppose, by contradiction, that this is not the case. Then, there exists a sequence \(\varPhi _n\) such that \(\varPhi _n (0)=0,\) \(\varPhi _n ^{'} (0)=\mathrm{Id}\) satisfying (4) and not the conclusion of the theorem.

If \((1-|z|)||\varPhi _n^{'} (z)|| \le A,\) with A constant, then \(\varPhi _n\) is normal, and we can assume \(\varPhi _n \rightarrow \varPsi ,\) with \(\varPsi (0)=0\) and \(\varPsi ^{'} (0)=\mathrm{Id}.\)

On an appropriate sphere \(\partial B (0,r),\) we get \(|\varPhi _n -\varPsi | < |\varPsi |.\) So by Rouche’s theorem and by the fact that the image of \(\varPsi \) contains a ball of radius R, we get that eventually also the images of \(\varPhi _n\) contain a ball of radius smaller or equal than R because we are interested in the inequality

$$\begin{aligned} |\varPhi _n - b - \varPsi +b| < |\varPsi - b| \le |\varPsi | + |b|, \end{aligned}$$

for all \(b \in B(a, R)\) and this will be satisfied possibly on a \(\partial B (0, r^{'})\) with \(r^{'} \le r\) which is a contradiction.

Hence, we can assume that \(\sup \nolimits _{|z| <1} (1-|z|) ||\varPhi _n^{'} (z)|| =\lambda _n \rightarrow + \infty .\)

As in the previous result, we can reparametrize and get

$$\begin{aligned} \varPsi _n := \varPhi _n (a_n + A_n z) \rightarrow \varPsi , \,\,\,\, \varPsi ^{'} (0)=\mathrm{Id}. \end{aligned}$$

Then, \(\varPsi ({\mathbb {B}}^k)\) contains a ball centered at \(\varPsi (0)=a,\,B(\varPsi (0), R).\)

Since by Rouche’s theorem:

$$\begin{aligned} |\varPsi _n - \varPsi | < |\varPsi | \end{aligned}$$

on \(\partial B(0,r),\) then eventually the image of \(\varPsi _n\) contains a ball of radius smaller or equal than R because we are interested in the inequality

$$\begin{aligned} |\varPsi _n - b - \varPsi +b| < |\varPsi - b| \le |\varPsi | + |b|, \end{aligned}$$

for all \(b \in B(a, R)\) and this will be satisfied possibly on a \(\partial B (0, r^{'})\) with \(r^{'} \le r.\)

Then, the image of \(\varPhi _n\) will contain eventually a ball of radius \(R^{'} \le R\) and we get the domain U.

The supremum of the set of positive numbers \(R^{'}\) such that \(\varPhi ({\mathbb {B}}^k)\) contains a ball of radius \(R^{'}\) is the so-called Landau’s number \(l(\varPhi ).\) The constant \(\rho \) is \(\inf \nolimits _{\varPhi \in {\mathcal {G}}} l(\varPhi )\) where \({\mathcal {G}}\) is the set of \(\varPhi :{\mathbb {B}}^k \rightarrow {\mathbb {C}}^k,\) \(\varPhi (0)=0,\) and \(\varPhi ^{'} (0)= \mathrm{Id}\) satisfying condition (4). \(\square \)

We point out that, following the same arguments of [14], it is possible to find an open set U on which the entire family of \(\varPhi \)’s satisfying (4) is injective, and it is also possible to relate the constant C with the uniform radius \(\rho .\)

Example 3.2

Consider the following family of hénon maps of \({\mathbb {C}}^2:\)

$$\begin{aligned} h_b(z,w)=(z^2+bw, z) \end{aligned}$$

with \(\delta < |b| \le \eta ,\) for \(\delta , \,\, \eta \) fixed constants.

Let \(f(z,w)=(e^{cz}-1,e^{cw}-1),\) with |c| small enough if you need f invertible, then the family

$$\begin{aligned} g_{b,c} (z,w) = \left\{ h_b \circ (e^{cz}-1, e^{cw}-1) \right\} \end{aligned}$$

satisfies the hypothesis of Theorem 3.1, up to dilation.

Furthermore, if the ball centered in the origin is enough small, then the ball of universal radius contained in the images of the family is a ball on which the maps of the family are invertible.

Corollary 3.3

Let \(f:{\mathbb {C}}^k \rightarrow {\mathbb {C}}^k\) be an entire map. Suppose that \(f^{'} (0) =\mathrm{Id}\) and suppose that there exists a constant C such that \(||f^{'} (z)|| \cdot ||f^{'} (z)^{-1}|| \le C,\) then \(f({\mathbb {C}}^k)\) contains balls of arbitrary large radius.

Proof

Indeed, \(\frac{1}{R}f(R \cdot B(0,1))\supset B(a,r)\) for any \(R>0,\) by Theorem 3.1. \(\square \)