Applications of Zalcman's lemma in $C^n$

Theorem 1.1. (Hurwitz's theorem) Let Ω be an open connected set in C ⁿ and {gj(z)} a sequence of holomorphic functions on Ω, converging uniformly on compact sets to a holomorphic function g. Then if gj(z) 6= 0 for all j, and all z, and g is nonconstant, we have g(z) 6= 0 for all z ∈ Ω. The theorem of Montel is easy consequence of Zalcman's lemma [10]. 1 ²⁰²⁰ Mathematics Subject Classification. 32A19. Key words and phrases. Zalcman's Lemma, normal families, holomorphic functions of several complex variables.

Theorem 1.2. (Montel's theorem) Let $\mathcal{F}$ be a family of holomorphic functions on an open set $\Omega \subseteq C^n$ that omit two fixed complex values. Then, each sequence of functions in $\mathcal{F}$ has a subsequence which converges uniformly on compact subsets.

Theorem 1.3. (Schotthy's theorem). Let M > 0 and $r \in [0,1)$ be given. Then there exists a constant C > 0 depending only on r and f(0) such that the following implication holds: If f is holomorphic in the unit ball B, omits 0 and 1 from its range, and if |f(0)| < M, then |f(z)| < C for all $z \in B(0,r) = z \in C^n : |z| < r$ .

Theorem 1.4. (Carathéodory's theorem) Let $\mathcal{F}$ be a family of functions holomorphic on $\Omega \subset C^n$ . Suppose that for some $\varepsilon > 0$ , there exist for each $f \in \mathcal{F}$ distinct points $a_f, b_f \in C$ such that for all $z \in \Omega$ , $f(z) \neq a_f, b_f$ and $$s(a_f, \infty)s(a_f, b_f)s(\infty, b_f) > \varepsilon$$ . Then $\mathcal{F}$ is normal on $\Omega$ .

Theorem 1.5. (Fatou's theorem) Let a(z) and b(z) be functions holomorphic on Ω ⊂ C ⁿ such that a(z) 6= b(z) for each z ∈ Ω. Let F be a family of functions holomorphic on Ω such that for each z ∈ Ω $$f(z) \neq a(z)$$ $f(z) \neq b(z)$ for all f ∈ F. Then F is normal on Ω.

Theorem 1.7. (R. Nevanlinna) [7, Theorem 17.3.10., p. 274] Let g be an entire function. Then g has at most two totally ramified (finite) values. Let f be a holomorphic function on an open connected set Ω in C ⁿ. Define the value set by $$A_f(a) = \{z \in \Omega : f(z) = a\} = f^{-1}[\{a\}].$$

Theorem 1.8. (Hurwitz's theorem). [12, Corollary p.80] Let Ω be an open connected set in C ⁿ and let {fj} be a sequence of holomorphic functions on Ω, converging uniformly on compact sets to a nonconstant holomorphic function f. If Af^j (a) = ∅ for all j then A^f (a) = ∅. Let ζ, v ∈ C ⁿ, v 6= 0. The set $$\{\xi \in C^n : \xi = \zeta + \lambda v, \lambda \in C\}$$ is called a complex line in C n. The restriction of an entire (holomorphic in C ⁿ) function g to a complex line {ξ = ζ + λv, λ ∈ C} clearly is an entire g(ζ + λ · v) of complex variable λ in C. Mimic the proof given in the one-dimensional case in [13, Theorem p. 219] we can prove the following normality criterion. The original result due to Lappan [9] corresponds to normal functions.

Theorem 1.9. family $\mathcal{F}$ of holomorphic functions on a domain $\Omega \subset C^n$ is normal on $\Omega$ if and only if for each compact set $K \subset \Omega$ , there exists a set $E = E(K) \subset C$ containing at least three distinct values and a finite constant M = M(K) > 0 for which $$(1.1) f^{\sharp}(z) \le M \quad z \in K, f(z) \in E$$ for all $f \in \mathcal{F}$ .

Definition 1.6. Let g(λ) be an entire (holomorphic in C) function, if the equation g(λ) = a, a ∈ C, has no simple roots then a called a totally ramified values. Note that an omitted value trivially satisfies this definition, but that it will be useful to distinguish between omitted values and non-omitted totally ramified values.