A note on a famous theorem of Pang and Zalcman

Theorem 1. Let $\mathcal{F}$ be a family of functions holomorphic on a domain D, all of whose zeros are of multiplicity (at least) 2. If there exist a non-zero constant b and a positive constant b such that for every $f \in \mathcal{F}$ , - (1) $f(z) = 0 \Rightarrow f''(z) = b$ , - (2) $f''(z) = b \Rightarrow 0 < |f'''(z)| \le M$ and - (3) $f'^{2}(z) = Bf(z)$ whenever $z \in \overline{E}_{f''}(b)$ , where B is a non-constant, then $\mathcal{F}$ is normal in D. Remark 1. Here, if f omits a constant b, we can say that all the zeros of f - b are of multiplicity $\infty$ . Remark 2. For the special cases that $\mathcal{F}$ is holomorphic functions and k=2 of Theorem A, from $\overline{E}_f(0) = \overline{E}_{f''}(b)$ , it is easy to deduce $\mathcal{F}$ satisfies the condition (3) of Theorem 1. Thus, in some sense, our result is an improvement of Theorem A. Meanwhile, we know that the condition $\overline{E}_f(0) = \overline{E}_{f^{(k)}}(b)$ is not necessary for holomorphic functions in Theorem A. Remark 3. We give an example to show that there exists a normal family $\mathcal{F}$ satisfying the conditions of Theorem 1. Consider the family $\mathcal{F} = \{f_n, n = 1, 2, \ldots\}$ on the unit disc, where $$f_n(z) = e^{\frac{z}{n}},$$ so that $$f'_n(z) = \frac{1}{n} e^{\frac{z}{n}}$$ and $f''_n(z) = \frac{1}{n^2} e^{\frac{z}{n}}$ . Let b be a non-zero constant and B = b. Then, it is easy to see the family $\mathcal{F}$ satisfies the conditions of Theorem 1 and $\mathcal{F}$ is normal on the unit disc. Remark 4. The assumption $0 < |f''(z)| \le M$ cannot be replaced by $|f''(z)| \le M$ . We have a counter-example [11] to show it. Consider the family $\mathcal{F} = \{f_n, n = 1, 2, \ldots\}$ on the unit disc, where $$f_n(z) = \frac{1}{n^2}(e^{nz} + e^{-nz} - 2) = \frac{1}{n^2}e^{-nz}(e^{nz} - 1)^2,$$ so that $$f_n^{(j)}(z) = n^{(j-2)}[e^{nz} + (-1)^j e^{-nz}], \quad j = 1, 2, \dots$$ It is easy to see all the zeros of $f_n$ are of multiplicity 2 and $$f_n(z) = 0 \Leftrightarrow f_n''(z) = 2 \Rightarrow f_n'''(z) = 0.$$ While the family $\mathcal{F}$ is not normal on the unit disc.

Lemma 1. Let $\mathcal{F}$ be a family of functions holomorphic on the unit disc, all of whose zeros have multiplicity at least k, and suppose that there exists $A \geq 1$ such that $|f^{(k)}(z)| \leq A$ whenever f(z) = 0, if $\mathcal{F}$ is not normal, then there exist, for each $0 \leq \alpha \leq k$ , - (a) a number 0 < r < 1; - (b) points $z_n$ , $z_n < r$ ; - (c) functions $f_n \in \mathcal{F}$ , and - (d) positive number $\rho_n \to 0$ such that $\rho_n^{-\alpha} f_n(z_n + \rho_n \xi) = g_n(\xi) \to g(\xi)$ locally uniformly, where g is a nonconstant holomorphic function on $\mathbb{C}$ , whose zeros have multiplicity at least k, such that $g^{\sharp}(\xi) \leq g^{\sharp}(0) = A + 1$ and $\rho(g) \leq 1$ . Here, as usual, $g^{\sharp}(\xi) = \frac{|g'(\xi)|}{1+|g(\xi)|^2}$ is the spherical derivative and $\rho(g)$ is the order of g. Next, we need to introduce a result, see [5, Theorem 4.1] or [10], which plays an important part in the proof of our Theorem.

Lemma 2. Let f be an entire function of order at most 1 and k be a positive integer, then $$m(r, \frac{f^{(k)}}{f}) = o(\log r), \quad as \quad r \to \infty.$$ Finally, we recall the theorem of Chang, Fang and Zalcman, see [3], which is crucial to the proof of our theorem.

Lemma 3. Let g be a non-constant entire function with $\rho(g) \leq 1$ , let $k \geq 2$ be an integer, and let a be a non-zero finite value. If $g(z) = 0 \Rightarrow g'(z) = a$ , and $g'(z) = a \Rightarrow g^{(k)}(z) = 0$ , then $$g(z) = a(z - z_0),$$ where $z_0$ is a constant.