Creating Limit Functions By The Pang-Zalcman Lemma

Theorem 1. Let $f(z) = R(z)e^{P(z)}$ be as in (1.4), where $P(z) = a_k(z - \alpha_1)...(z - \alpha_k)$ (the $\alpha_i$ 's may occur with repititions), $a_k \neq 0$ ; $R(z) = \frac{P_1(z)}{P_2(z)}$ where $P_1(z) = (z - \gamma_1)^{l_1}...(z - \gamma_m)^{l_m}$ , $P_2(z) = (z - \beta_1)^{j_1}...(z - \beta_l)^{j_l}$ . We assume that $\gamma_1, \dots, \gamma_m$ ; $\beta_1, \dots, \beta_l$ are all distinct. Let $L_1 := |P_1| = l_1 + ... + l_m$ , $L_2 := |P_2| = j_1 + ... + j_l$ . Then for the various values of $-1 < \alpha < 1$ , $\Pi_{\alpha}(f)$ is given as follows: I. $$k = |P| = 1$$ If $\alpha = 0$ , then $$\Pi_0(f) = \left\{ k_0 e^{A_1 \zeta} : k_0 \neq 0, \arg A_1 = \arg a_1 \right\} \bigcup \left\{ f(C_1 + C_2 \zeta) : C_1 \in \mathbb{C}, C_2 > 0 \right\}.$$ If $$0 < \alpha < 1$$ , then $$\Pi_{\alpha}(f) = \left\{ k_0 e^{A_1 \zeta} : k_0 \neq 0, \arg A_1 = \arg a_1 \right\} \bigcup \\ \left\{ e^{P(\gamma_i)} \tilde{R}_{\gamma_i}(\gamma_i) (A_1 \zeta + A_0)^{l_i} : 1 \leq i \leq m, A_0 \in \mathbb{C}, A_1 > 0 \right\}.$$ If $$-1 < \alpha < 0$$ , then $$\Pi_{\alpha}(f) = \left\{ k_0 e^{A_1 \zeta} : k_0 \neq 0, \arg A_1 = \arg a_1 \right\} \bigcup \left\{ e^{P(\beta_i)} \hat{R}_{\beta_i}(\beta_i) (A_1 \zeta + A_0)^{-j_i} : 1 \leq i \leq l, A_0 \in \mathbb{C}, \ A_1 > 0 \right\}.$$ II. $$k \geq 2$$ If $\alpha = 0$ , then $$\Pi_0(f) = \left\{ f(C_1 + C_2 \zeta) : C_1 \in \mathbb{C}, C_2 > 0 \right\} \bigcup \left[ \bigcup_{l=0}^{k-1} \left\{ e^{A_1 \zeta + A_0} : A_0 \in \mathbb{C}, \arg A_1 = \left( \pm \frac{\pi}{2} (k-1) + \arg a_k + (k-1) 2\pi l \right) / k \right\} \right].$$ If $$0 < \alpha < 1$$ , then for $$k=2$$ $$\Pi_{\alpha}(f) = \left[ \bigcup_{i=1}^{m} \left\{ e^{P(\gamma_i)} A(\zeta + C)^{l_i} : \arg A = \arg \tilde{R}_{\gamma_i}(\gamma_i), C \in \mathbb{C} \right\} \right] \bigcup \left\{ e^{A_0 + A_1 \zeta} : A_0 \in \mathbb{C}, \frac{\pi}{4} + \frac{\arg a_2}{2} \le \arg A_1 \le \frac{3\pi}{4} + \frac{\arg a_2}{2} \quad or \\ \frac{5\pi}{4} + \frac{a_2}{2} \le \arg A_1 \le \frac{7\pi}{4} + \frac{\arg a_2}{2} \right\} .$$ For $$k \geq 3$$ $$\Pi_{\alpha}(f) = \left[ \bigcup_{i=1}^{m} \left\{ e^{P(\gamma_i)} A(\zeta + C)^{l_i} : \arg A = \arg \tilde{R}_{\gamma_i}(\gamma_i), C \in \mathbb{C} \right\} \right] \bigcup \left\{ e^{A_1 \zeta + A_0} : A_0 \in \mathbb{C}, A_1 \neq 0 \right\} .$$ If $-1 < \alpha < 0$ , then for $$k=2$$ $$\Pi_{\alpha}(f) = \left[ \bigcup_{i=1}^{l} \left\{ e^{P(\beta_i)} A(\zeta + C)^{-j_i} : \arg A = \arg \hat{R}_{\beta_i}(\beta_i), C \in \mathbb{C} \right\} \right] \bigcup \left\{ e^{A_0 + A_1 \zeta} : A_0 \in \mathbb{C}, -\frac{\pi}{4} + \frac{\arg a_2}{2} \le \arg A_1 \le \frac{\pi}{4} + \frac{\arg a_2}{2} \right\} \\ or \quad \frac{3\pi}{4} + \frac{\arg a_2}{2} \le \arg A_1 \le \frac{5\pi}{4} + \frac{\arg a_2}{2} \right\} .$$ For k > 3 $$\Pi_{\alpha}(f) = \left[ \bigcup_{i=1}^{l} \left\{ e^{P(\beta_i)} A(\zeta + C)^{-j_i} : \arg A = \arg \hat{R}_i(\beta_i), C \in \mathbb{C} \right\} \right] \bigcup \left\{ e^{A_0 + A_1 \zeta} : A_0 \in \mathbb{C}, A_1 \neq 0 \right\} .$$ Observe that in each of the three intervals $\alpha = 0$ , $0 < \alpha < 1$ and $-1 < \alpha < 0$ , $\Pi_{\alpha}(f)$ is independent of $\alpha$ . The proof of Theorem 1 is similar to climbing a ladder with four steps where each step is more complicated then the former step. In the first step we calculate Πα(M) for a general monome, M(z) = (z − α) k . In the second step we find Πα(P) where P is a general nonconstant polynomial. In step 3 we calculate Πα(R), where R is a general nonconstant rational function, and finally in the fourth step we find Πα(Re^P ). In each step we rely on the results of the previous steps. The first three steps is the contents of section 2, the proof of Theorem 1 is actually the fourth step which we prove in section 3. We note that for a nonconstant rational function, z⁰ = 0 is the only point of non-normality in C, and this is the situation in the first three steps. For f = Re^P , the points of non-normality lies on few rays through the origin, as we will see in the sequel. Throughout the proof we often deal with the connections between {zn} and {ρn} in the LPZ Lemma. We hope this will contribute to the better understanding of the potential of this somewhat obscure lemma. As it is always possible to move to convergent subsequences (in the extended sense), we shall always assume without loss of generality that the sequences {knzn}, {knρn} from [\(1.3\)](#page-3-1) converge (in the extended sense). This assumption also applies to other sequences of complex numbers involved in our calculations. The importance of this paper, beyond the result obtained in Theorem 1, lies in the technique that we used. The possible connections between zⁿ and ρⁿ in [\(1.1\)](#page-2-0) were used to deduce the limit function g. We note that the Pang-Zalcman Lemma is a common tool to establish normality of families of meromorphic functions. However, the proof of this lemma does not give an explicit relation between zⁿ to ρn, because some unknown parameter is involved in this relation (see [\[8,](#page-31-2) Lemma 2], [\[9,](#page-31-3) Theorem 1]). Hence, in general there is some difficulty in determing the limit function g. We expect that the detailed calculation that given here will contribute and promote the study of this subject. - <span id="page-7-2"></span>2. Calculating $\Pi_{\alpha}(M)$ , $\Pi_{\alpha}(P)$ and $\Pi_{\alpha}(R)$ - <span id="page-7-1"></span><span id="page-7-0"></span>2.1. First step: Calculating $\Pi_{\alpha}(M)$ where $M(z) = (z - \beta)^k$ . Let $-1 < \alpha < 1$ and assume that $M_{n,\alpha}(\zeta) \Rightarrow g(\zeta)$ , (where g is a nonconstant entire function). This means that $$(2.1) (k_n \rho_n^{1-\frac{\alpha}{k}} \zeta + \frac{k_n z_n - \beta}{\rho_n^{\frac{\alpha}{k}}})^k \Rightarrow g(\zeta) .$$ The left hand side of (2.1) has a single zero of multiplicity k in $\mathbb{C}$ , and thus, it follows by Rouché's Theorem that $g(\zeta)$ is also a monome of degree k. There must be $0 < A < \infty$ and $C \in \mathbb{C}$ , such that $k_n \rho_n^{1-\frac{\alpha}{k}} \to A$ and $\frac{k_n z_n - \beta}{\rho_n^{\frac{\alpha}{k}}} \to C$ and so $g(\zeta) = (A\zeta + C)^k$ . Conversely, given A > 0 and $C \in \mathbb{C}$ , we set <span id="page-7-3"></span>(2.2) $$k_n = n, \quad \rho_n = \left(\frac{A}{n}\right)^{\frac{k}{k-\alpha}}, \quad z_n = \frac{A^{\frac{\alpha}{k-\alpha}}C + \beta n^{\frac{\alpha}{k-\alpha}}}{n^{1+\frac{\alpha}{k-\alpha}}}$$ to get (for every n) $M_n(\zeta) = (A\zeta + C)^k$ . Thus, for every $-1 < \alpha < 1$ <span id="page-7-4"></span>(2.3) $$\Pi_{\alpha}(M) = \{ (A\zeta + C)^k : A > 0, C \in \mathbb{C} \} .$$ <span id="page-8-0"></span>2.2. Second step: Calculating $\Pi_{\alpha}(P)$ for a nonconstant polynomial P(z). Let $P(z) = L(z - \gamma_1)^{l_1}...(z - \gamma_m)^{l_m}, \ \gamma_i \neq \gamma_j, \ i \neq j,$ $k := l_1 + l_2... + l_m$ . Assume first that $\alpha = 0$ and that <span id="page-8-1"></span> $$(2.4) P_{n,0}(\zeta) = P(k_n \rho_n \zeta + k_n z_n) \Rightarrow g(\zeta) .$$ By substituting $\zeta = 0$ in (2.4), we get that $\{k_n z_n\}$ is bounded and thus $k_n z_n \to C \in \mathbb{C}$ (recall that we always assume without loss of generality that $\{k_n z_n\}$ , $\{k_n \rho_n\}$ , etc. converge). Now, if $k_n \rho_n \to 0$ then g is constant and in case that $k_n \rho_n \to \infty$ then $g(\zeta) = \infty$ for every $\zeta \neq 0$ . Hence $k_n \rho_n \to A$ , $0 < A < \infty$ and we have $g(\zeta) = P(A\zeta + C)$ . On the other hand, given $0 < A < \infty$ and $C \in \mathbb{C}$ , the trivial setting $k_n = n$ , $\rho_n = \frac{A}{n}$ , $z_n = \frac{C}{n}$ gives $P_{n,0}(\zeta) = P(A\zeta + C)$ and we get <span id="page-8-3"></span>(2.5) $$\Pi_0(P) = \{ P(A\zeta + C) : A > 0, C \in \mathbb{C} \}.$$ Consider now the case where $0 < \alpha < 1$ . Here $P_{n,\alpha}(\zeta) \Rightarrow g(\zeta)$ means <span id="page-8-2"></span> $$(2.6) \qquad \frac{L(k_n\rho_n\zeta + k_nz_n - \gamma_1)^{l_1}...(k_n\rho_n\zeta + k_nz_n - \gamma_m)^{l_m}}{\rho_n^{\alpha}} \Rightarrow g(\zeta) .$$ Because of $\rho_n^{\alpha} \to 0$ , then by substituting $\zeta = 0$ in (2.6), we get that there exists $1 \le i \le m$ such that $k_n z_n \to \gamma_i$ , since otherwise $P_{n,\alpha}(0) \to \infty$ , and this would be a contradiction. Without loss of generality, we assume that i=1.

Lemma 3.1. Let f be a nonconstant meromorphic function in $\mathbb{C}$ and $-1 < \alpha < 1$ . Then - (1) If $g(\zeta) \in \Pi_{\alpha}(f)$ then for every $C \in \mathbb{C}$ $g(\zeta + C) \in \Pi_{\alpha}(f)$ and - (2) If $e^{a\zeta+b} \in \Pi_{\alpha}(f)$ then for every $a' \neq 0$ such that $\arg(a') = \arg(a)$ and for every $b' \in \mathbb{C}$ , $e^{a'\zeta+b'} \in \Pi_{\alpha}(f)$ . Proof. Suppose that $g \in \Pi_{\alpha}(f)$ , then we have $\frac{f(k_n z_n + k_n \rho_n(\zeta + C))}{\rho_n^{\alpha}} \stackrel{\chi}{\Rightarrow} g(\zeta)$ in $\mathbb{C}$ , with $\rho_n \to 0^+$ , $z_n \to z_0$ and $k_n \in \mathbb{N}$ . We set $\rho'_n = \rho_n$ , $z'_n = z_n + \rho_n C \to z_0$ and get $$\frac{f(k_n z_n' + k_n \rho_n' \zeta)}{\rho_n'^{\alpha}} = \frac{f(k_n z_n + k_n \rho_n(\zeta + C))}{\rho_n^{\alpha}} \stackrel{\chi}{\Rightarrow} g(\zeta + C) ,$$ and this proves (1). For the proof of (2) assume that $\frac{f(k_n z_n + k_n \rho_n \zeta)}{\rho_n^{\alpha}} \stackrel{\chi}{\Rightarrow} e^{a\zeta + b}$ in $\mathbb{C}$ . Define for a' with $\arg a' = \arg a$ , $\rho'_n = \frac{a'}{a}\rho_n \to 0^+$ and $(\frac{A'}{A})^{-\alpha} = e^{b_0}$ , where $b_0 \in \mathbb{R}$ . We have $$\frac{f(k_n z_n + k_n \rho'_n \zeta)}{(\rho'_n)^{\alpha}} = \frac{f(k_n z_n + k_n \rho_n(\frac{a'}{a}\zeta))}{\rho_n^{\alpha}(a'/a)^{\alpha}} \stackrel{\chi}{\Rightarrow} g(\frac{a'}{a}\zeta)e^{b_0} = e^{a'\zeta + b + b_0}.$$ By (1) we can replace $b + b_0$ with every $b' \in \mathbb{C}$ . This completes the proof of the lemma. Remark. Let F be a family of non-vanishing holomorphic functions which is not normal at $z_0$ and let $-1 < \alpha < 1$ . Then the convergence process (1.1) in the LPZ Lemma guarantees a limit function $g(\zeta)$ with $g^{\#}(\zeta) \leq 1$ for every $\zeta \in \mathbb{C}$ . By a theorem of Clunie and Hayman [1, Theorem 3], the order of g is at most 1 and since $g(\zeta) \neq 0$ , $\zeta \in \mathbb{C}$ , by Hurwitz's Theorem we deduce that $g(\zeta) = e^{a\zeta+b}$ . The results which we will prove in the detailed process of calculating $\Pi_{\alpha}(Re^P)$ are indeed consistent with this theorem of Clunie and Hayman.

Lemma 3.2. Let $f = Re^P$ be given by (3.1). Then the points where F(f) is not normal in $\mathbb{C}$ are exactly <span id="page-15-0"></span>(3.2) $$\left\{ \bigcup_{l=0}^{k-1} R_{\theta_k^+(l)} \right\} \bigcup \left\{ \bigcup_{l=0}^{k-1} R_{\theta_k^-(l)} \right\}$$ where for every $0 \le l \le k-1$ , $\theta_k^+(l)$ and $\theta_k^-(l)$ are defined by $\theta_k^{\pm}(l) = \frac{\pm \frac{\pi}{2} - \arg a_k}{k} + \frac{2\pi l}{k}$ and $\arg a_k$ is taken to be in $[0, 2\pi)$ . Observe that for every $0 \le l \ne j \le k - 1$ , $\theta_k^{\pm}(l) \ne \theta_j^{\pm}(l)$ .