An improvement of Zalcman's lemma in $C^n$

Theorem 1.1. (Marty's Criterion, see [2]) A family $\mathcal{F}$ of functions holomorphic on $\Omega$ is normal on $\Omega \subset C^n$ if and only if for each compact subset $K \subset \Omega$ there exists a constant M(K) such that at each point $z \in K$ $$(1.2) f^{\sharp}(z) < M(K)$$ for all $f \in \mathcal{F}$ . Marty's criterion is one of the more important results in function theory widely used for determining the normality of a family of holomorphic functions. Marty's criterion is one of the main ingredients of the proof of Zalcman's lemma [2]. We prove the following improved result of Zalcman-Pang's [3].

Theorem 1.2. Let $\mathcal{F}$ be a family of functions holomorphic on $\Omega \subset \mathbb{C}^n$ . Then $\mathcal{F}$ is not normal at some point $z_0 \in \Omega$ if and only if for each $\alpha \in (-1, \infty)$ there exist sequences $f_j \in \mathcal{F}$ , $z_j \to z_0$ , $r_j \to 0$ , such that the sequence $$g_j(z) := r_j^{\alpha} f_j(z_j + r_j z)$$ 1 $^1991\ Mathematics\ Subject\ Classification.\ 32A19.$ Key words and phrases. Zalcman's Lemma; Zalcman-Pang's Lemma; Normal families; Holomorphic functions of several complex variables. converges locally uniformly in $C^n$ to a non-constant entire function g satisfying $g^{\sharp}(z) \leq g^{\sharp}(0) = 1$ . In case n=1 this theorem was proved in Hua [6, Lemma 6]. A similar result was proved by Chen and Gu [1, Th.2] (see also Xue and Pung [8], cf. Hua [6]). The special case $\alpha=0$ of Theorem 1.2 was proved in Zalcman [9, p. 814] and is known as Zalcman's rescaling lemma. Zalcman's lemma - now upgraded to the status of theorem - was first stated at [9]; for a state-of-the-art version, see [10, Lemma 2]. The plan of this paper is as follows. In Section 2, we state and prove a number of auxiliary results, some of which are of independent interest. In Section 3, we give the proof of main theorem. In Section 4, we give two applications of main theorem.

Theorem 2.1. (Hurwitz's theorem, see, e.g. [7, (1.5.16) Lemma, p. 24]) Let $\Omega$ be a domain of $C^n$ and $\{h_j\}$ a sequence of non-vanishing holomorphic functions $h_j$ which converges uniformly on compact subsets to a holomorphic function h on $\Omega$ . Then h vanishes either everywhere or nowhere. Note that $$L_z(\log(1+|f(z)|^2),v) = \frac{|(Df(z),v)|^2}{(1+|f(z)|^2)^2}$$ on $\Omega$ . Appealing to the Cauchy-Schwarz inequality it is easy to show that $$(1+|f(z)|^2)f^{\sharp}(z) = |Df(z)|.$$ The following lemma will play a crucial role in the proof of Theorem 1.2.

Lemma 2.2. Let f be a holomorphic function on the closed unit ball $\overline{B(0,1)}$ , and $\alpha$ be a real number with $-1 < \alpha < \infty$ . Suppose $$\max_{|z| \leq 1/j} \frac{(1-j|z|)^{1+\alpha}(1+|f(z)|^2)f^{\sharp}(z)}{1+(1-j|z|)^{2\alpha}|f(z)|^2} > 1.$$ Then there exists a point $\xi$ , $|\xi^| < 1/j$ , and a real number $\rho$ , $0 < \rho < 1$ , such that $$\max_{|z| \le 1/j} \frac{(1-j|z|)^{1+\alpha} \rho^{1+\alpha} (1+|f(z)|^2) f^{\sharp}(z)}{1+(1-j|z|)^{2\alpha} \rho^{2\alpha} |f(z)|^2} = \frac{(1-j|\xi^|)^{1+\alpha} \rho^{1+\alpha} (1+|f(\xi^)|^2) f^{\sharp}(\xi^)}{1+(1-j|\xi^|)^{2\alpha} \rho^{2\alpha} |f(\xi^*)|^2} = 1.$$ Proof. Set (2.1) $$\varphi(t,z) := \frac{(1-j|z|)^{1+\alpha}\rho^{1+\alpha}(1+|f(z)|^2)f^{\sharp}(z)}{1+(1-j|z|)^{2\alpha}\rho^{2\alpha}|f(z)|^2}.$$ Suppose that $\varphi(1, z_1^*) := \max_{|z| \le 1/j} \varphi(1, z) > 1$ . Since $(1 + |f(z)|^2) f^{\sharp}(z)$ is bounded on $|z| \le 1/j$ we have <span id="page-1-1"></span>(2.2) $$\varphi(t,z) \le (1-j|z|)^{1-\alpha}t^{1-\alpha}M.$$ It follows $\varphi(t,z)$ is continuous on $[0,1] \times \{z \in C^n : |z| \le 1/j\}$ and $\varphi(0,z) = 0$ on $\{z \in C^n : |z| \le 1/j\}$ . Hence $\varphi(0, z_1^) = 0$ and $\varphi(1, z_1^) > 1$ . By continuity of $\varphi(t, z)$ on $[0, 1] \times \{z \in C^n : |z| \le 1/j\}$ , there exists $\rho_1$ , $0 < \rho_1 < 1$ , such that $\varphi(\rho_1, z_1^*) = 1$ . Repeating this procedure we can find $\rho_m$ , $0 < \rho_m < 1$ , and $z_m$ , $|z_m^| < 1/j$ , such that (2.3) $$\max_{|z| \le 1/j} \varphi(\rho_1 \dots \rho_m, z) = \varphi(\rho_1 \dots \rho_m, z_m^*) > 1.$$ $$\varphi(\rho_1 \dots \rho_m \rho_{m+1}, z_m^*) = 1.$$ The sequence $\{x_m := \rho_1 \dots \rho_m\}$ is a bounded and decreasing sequence. Then the greatest lower bound of the set $\{x_m : m \in N\}$ , say $\rho$ , is the limit of $\{x_m\}$ . The sequence $\{z_m^\}$ contains a subsequence, again denoted by $\{z_m^\}$ , such that $\lim_{m\to\infty} z_m^ = \xi$ . From (2.2) follows that $0 < \rho < 1$ and $|\xi^*| < 1/j$ . <span id="page-2-3"></span>We claim that (2.4) $$\max_{|z| \le 1/j} \lim_{m \to \infty} \varphi(\rho_1 \dots \rho_m, z) = \lim_{m \to \infty} \max_{|z| \le 1/j} \varphi(\rho_1 \dots \rho_m, z).$$ Since $\varphi$ is continuous function on $[0,1] \times B(0,1/j)$ by the Weierstrass theorem (see [4, Theorem (Weierstrass) p. 565]) we can find $|\eta| < 1/j$ and $|w_m| < 1/j$ such that <span id="page-2-1"></span>(2.5) $$\max_{|z| \le 1/j} \lim_{m \to \infty} \varphi(\rho_1 \dots \rho_m, z) = \max_{|z| \le 1/j} \varphi(\rho, z) = \varphi(\rho, \eta);$$ <span id="page-2-2"></span> $$(2.6) \quad \varphi(\rho_1 \dots \rho_m, \eta) \le \max_{|z| \le 1/j} \varphi(\rho_1 \dots \rho_m, z) = \varphi(\rho_1 \dots \rho_m, w_m), \quad m = 1, 2, \dots$$ By the Bolzano-Weierstrass theorem there is an infinite subsequence of $\{w_m\}$ , again denoted by $\{w_m\}$ , and $\varsigma$ , $|\varsigma| \leq 1/j$ , such that $w_m \to \varsigma$ as $m \to \infty$ . Because $w_m \to \varsigma$ and $\rho_1 \dots \rho_m \to \rho$ as $m \to \infty$ and $\varphi$ is continuous function on $[0,1] \times \overline{B(0,1/j)}$ from (2.5) and (2.6) we see $$\varphi(\rho,\eta) \le \lim_{m \to \infty} \max_{|z| < 1/j} \varphi(\rho_1 \dots \rho_m, z) = \varphi(\rho,\varsigma) \le$$ $$\max_{|z| \le 1/j} \varphi(\rho, z) = \max_{|z| \le 1/j} \lim_{m \to \infty} \varphi(\rho_1 \dots \rho_m, z) = \varphi(\rho, \eta).$$ That is, the claim (2.4) is proved. Combining (2.4) and (2.5) we obtain $$\max_{|z| \le 1/j} \varphi(\rho, z) = \varphi(\rho, \xi^) = 1 \ (|\xi^| < 1/j).$$ <span id="page-2-0"></span>The proof of the lemma is complete.

Theorem 4.1. Let some $\varepsilon > 0$ be given and set $$\mathcal{F} = \{f \text{ holomorphic in } \Omega : f^{\sharp}(z) > \varepsilon \text{ for all } z \in \Omega\}.$$ Then $\mathcal{F}$ is normal in $\Omega$ .

Theorem 4.2. Let $\mathcal{F}$ be a family of zero-free holomorphic functions in a domain $\Omega \subset C^n$ . The statement of Theorem 1.2 remains valid if $-1 \leq \alpha < \infty$ is replaced with $-\infty < \alpha < \infty$ . Proof of Theorem 4.2. It need to consider only the case $-\infty < \alpha < 0$ . Since a family $\{1/f, f \in \mathcal{F}\}$ conforms to the hypotheses of Theorem 4.2 the earlier argument shows that there exist sequences $1/f_j$ , $z_j \to z_0$ , $r_j \to 0$ , such that the sequence $$g_j(z) := \frac{r_j^{\alpha}}{f_j(z_j + r_j z)} \quad (0 \le \alpha < \infty \quad arbitrary)$$ converges locally uniformly in $C^n$ to a non-constant entire function g satisfying $g^{\sharp}(z) \leq g^{\sharp}(0) = 1$ . By Hurwitz's theorem either $g \equiv 0$ or g never vanishes. Since $g^{\sharp}(0) = 1$ it is easy to see that g never vanishes then 1/g is entire function in $C^n$ . It follows $r_j^{-\alpha} f_j \to 1/g$ uniformly in $C^n$ . Since Levi form vanishes for any pluriharmonic function, $L_z(\log(1+|1/g|^2),v) = L_z(\log(1+|g|^2),v) - 2L_z(\log|g|,v) = L_z(\log(1+|g|^2),v).$ Therefore, $$q^{\sharp}(z) = (1/q)^{\sharp}(z).$$ For every $z \in C^n$ we have $g^{\sharp}(z) \leq g^{\sharp}(0) = 1$ , hence $$(1/g)^{\sharp}(z) \le (1/g)^{\sharp}(0) = 1.$$ The case −∞ ≤ α < 0 is proved. This completes the proof of the theorem.