On The Relationship Between The Logarithmic Lower Order of Coefficients and The Growth of Solutions of Complex Linear Differential Equations in $\overline{\mathbb{C}}\setminus\{z_{0}\}$

Theorem 7. Let $A_0(z),...,A_{k-1}(z)$ be meromorphic functions in $\overline{\mathbb{C}}\setminus\{z_0\}$ of finite logarithmic order. Suppose there exists an integer s $(0 \le s \le k-1)$ such that $A_s(z)$ satisfies $\liminf_{r \longrightarrow 0} \frac{m_{z_0}(r,A_s)}{T_{z_0}(r,A_s)} = \delta > 0$ , $\max\left\{\sigma_{\log}(A_j,z_0): j \ne s\right\} \le \mu_{\log}(A_s,z_0) < +\infty$ and $$\sum_{\sigma_{\log}(A_j, z_0) = \mu_{\log}(A_s, z_0) \ge 1, j \ne s} \tau_{\log}(A_j, z_0) < \delta_{\underline{\tau}_{\log}}(A_s, z_0) < +\infty.$$ Then, every meromorphic solution $f(z)(\not\equiv 0)$ in $\overline{\mathbb{C}}\setminus\{z_0\}$ of (2) satisfies $0 \leq \mu_{\log}(A_s, z_0) - 1 \leq \mu_{\log}(f, z_0)$ and $\mu_{\log}(A_s, z_0) \leq \mu_{\log}(f, z_0)$ if $\mu_{\log}(A_s, z_0) > 1$ . Remark 3. We can also replace the conditions $\max\left\{\sigma_{\log}(A_j,z_0):j\neq s\right\}\leq \mu_{\log}(A_s,z_0)<+\infty$ and $$\sum_{\sigma_{\log}(A_j, z_0) = \mu_{\log}(A_s, z_0) \ge 1, j \ne s} \tau_{\log}(A_j, z_0) < \delta_{\underline{\tau}_{\log}}(A_s, z_0) < +\infty$$ in Theorem 7 by $\limsup_{r\longrightarrow 0} \frac{\sum_{j\neq s} m_{z_0}(r,A_j)}{m_{z_0}(r,A_s)} < 1$ or we replace the condition $\liminf_{r\longrightarrow 0} \frac{m_{z_0}(r,A_s)}{T_{z_0}(r,A_s)} = \delta > 0$ by $\lambda_{\log}(\frac{1}{A_s},z_0)+1 < \mu_{\log}(A_s,z_0)$ , which clearly includes the assumption that $\mu_{\log}(A_s,z_0)>1$ . Remark 4. The results in Theorem 6 and Theorem 7 may be understood as an extension respectively of Theorem 6 and Theorem 2 in [12] when an arbitrary coefficient $A_s$ dominating the others coefficients by its lower logarithmic order and lower logarithmic type.

Lemma 1. ([10, 20]). Let f(z) be a non-constant analytic function in $\overline{\mathbb{C}}\setminus\{z_0\}$ and let $V_{z_0}(r, f)$ be a central index of f(z) near the singular point $z_0$ . Then <span id="page-5-3"></span> $$\sigma_{[p,q]}(f,z_0) = \limsup_{r \to 0} \frac{\log_p^+ V_{z_0}(r,f)}{\log_q \frac{1}{r}}, \qquad \mu_{[p,q]}(f,z_0) = \liminf_{r \to 0} \frac{\log_p^+ V_{z_0}(r,f)}{\log_q \frac{1}{r}}.$$

Lemma 2. ([16]). Let f be a non-constant meromorphic function in $\overline{\mathbb{C}}\setminus\{z_0\}$ . Then there exists a set $E_1$ of (0,1) that has finite logarithmic measure such that for all j=0,...,k, we have $$\frac{f^{(j)}(z_r)}{f(z_r)} = \left(\frac{V_{z_0}(r,f)}{z_0 - z_r}\right)^j (1 + o(1)),$$ <span id="page-5-1"></span>as $r \to 0$ , $r \notin E_1$ , where $z_r$ is a point in the circle $|z - z_0| = r$ that satisfies $|f(z_r)| = \max_{|z - z_0| = r} |f(z)|$ .

Lemma 3. ([10, 20]). Let f be a non-constant analytic function in $\overline{\mathbb{C}}\setminus\{z_0\}$ with $\mu_{[p,q]}(f,z_0)\leq\sigma_{[p,q]}(f,z_0)<\infty$ . Then (i) there exists a set $E_2\subset (0,1)$ that has infinite logarithmic measure such that for all $|z-z_0|=r\in E_2$ , we have $$\sigma_{[p,q]}(f,z_0) = \lim_{r \to 0} \frac{\log_p T_{z_0}(r,f)}{\log_q \frac{1}{r}} = \lim_{r \to 0} \frac{\log_{p+1} M_{z_0}(r,f)}{\log_q \frac{1}{r}}.$$ (ii) there exists a set $E_3 \subset (0,1)$ that has infinite logarithmic measure such that for all $|z-z_0|=r \in E_3$ , we have $$\mu_{[p,q]}(f,z_0) = \lim_{r \to 0} \frac{\log_p T_{z_0}(r,f)}{\log_q \frac{1}{r}} = \lim_{r \to 0} \frac{\log_{p+1} M_{z_0}(r,f)}{\log_q \frac{1}{r}}.$$ By using similar proofs as for Lemma 3, we can prove the following lemma.

Lemma 4. Let f be a non-constant analytic function in $\overline{\mathbb{C}}\setminus\{z_0\}$ with $\mu=\mu_{\log}(f,z_0)\leq\sigma_{\log}(f,z_0)=\sigma$ . Then (i) there exists a set $E_4$ of (0,1) that has infinite logarithmic measure such that for all $|z-z_0|=r\in E_4$ , we have <span id="page-6-0"></span> $$\lim_{r \to 0} \frac{\log \log M_{z_0}(r, f)}{\log \log \frac{1}{r}} = \lim_{r \to 0} \frac{\log T_{z_0}(r, f)}{\log \log \frac{1}{r}} = \sigma.$$ (ii) there exists a set $E_5$ of (0,1) that has infinite logarithmic measure such that for all $|z-z_0|=r\in E_5$ , we have $$\lim_{r \to 0} \frac{\log \log M_{z_0}(r, f)}{\log \log \frac{1}{r}} = \lim_{r \to 0} \frac{\log T_{z_0}(r, f)}{\log \log \frac{1}{r}} = \mu.$$

Lemma 5. Let $A_0(z), ..., A_{k-1}(z)$ be analytic functions in $\overline{\mathbb{C}} \setminus \{z_0\}$ of finite logarithmic order. If there exists an integer $s(0 \le s \le k-1)$ such that $1 \le \max \{\mu_{\log}(A_s, z_0), \sigma_{\log}(A_j, z_0) : j \ne s\} \le \alpha$ , then every analytic solution $f(z) (\not\equiv 0)$ in $\overline{\mathbb{C}} \setminus \{z_0\}$ of (2) satisfies $\mu_{[2,2]}(f,z_0) \le \alpha$ .

Lemma 6. ([11]). Let $A_0(z), ..., A_{k-1}(z)$ be analytic functions in $\overline{\mathbb{C}}\setminus\{z_0\}$ of finite logarithmic order with $\max\{\sigma_{\log}(A_j, z_0): j=0,...,k-1\} \leq \beta < +\infty$ . Then, every analytic solution $f(z)(\not\equiv 0)$ in $\overline{\mathbb{C}}\setminus\{z_0\}$ of (2) satisfies $\sigma_{[2,2]}(f,z_0) \leq \beta$ .

Lemma 7. ([14]). Let f be a non-constant meromorphic function in $\overline{\mathbb{C}}\setminus\{z_0\}$ , let $\lambda>0$ , $\varepsilon>0$ be given real constants and $j\in\mathbb{N}$ . Then (i) there exist a set $E_6 \subset (0,1)$ of finite logarithmic measure and a constant C>0 that depends only on $\lambda$ and j such that for all $|z-z_0|=r\in (0,1)\backslash E_6$ , we have <span id="page-7-0"></span> $$\left| \frac{f^{(j)}(z)}{f(z)} \right| \le C \left[ \frac{1}{r^2} T_{z_0}(\lambda r, f) \log T_{z_0}(\lambda r, f) \right]^j. \tag{8}$$ (ii) there exist a set $E_7 \subset [0, 2\pi)$ that has a linear measure zero and a constant C > 0 that depends on $\lambda$ and j such that for all $\theta \in [0, 2\pi) \setminus E_7$ , there exists a constant $r_0 = r_0(\theta) > 0$ such that (8) holds for all z satisfying $\arg(z - z_0) \in [0, 2\pi) \setminus E_7$ and $r = |z - z_0| < r_0$ .

Lemma 8. Let $A_0(z),...,A_{k-1}(z)$ be analytic functions in $\overline{\mathbb{C}}\setminus\{z_0\}$ of finite logarithmic order with $\max\left\{\sigma_{\log}(A_j,z_0):j\neq 0\right\}<\sigma_{\log}(A_0,z_0)=\sigma<+\infty$ . Then, every analytic solution $f(z)(\not\equiv 0)$ in $\overline{\mathbb{C}}\setminus\{z_0\}$ of (2) satisfies $\sigma_{\log}(A_0,z_0)-1\leq \sigma_{\log}(f,z_0)\leq \sigma_{\log}(A_0,z_0)$ and $\sigma_{[2,2]}(f,z_0)=\sigma_{\log}(A_0,z_0)$ if $\sigma_{\log}(A_0,z_0)>1$ .

Lemma 9. ([6]). Let f be a non-constant meromorphic function in $\overline{\mathbb{C}}\setminus\{z_0\}$ and let $k\in\mathbb{N}$ . Then $$m_{z_0}\left(r, \frac{f^{(k)}(z)}{f(z)}\right) = O\left(\log T_{z_0}(r, f) + \log \frac{1}{r}\right), \quad \text{for all } r \in (0, 1) \setminus E_8 \quad \text{with } m_l(E_8) < \infty.$$ If f is of finite order, then $$m_{z_0}(r, \frac{f^{(k)}(z)}{f(z)}) = O(\log \frac{1}{r}), \qquad r \in (0, 1).$$

Lemma 10. ([20]). Let f be a non-constant meromorphic function in $\overline{\mathbb{C}}\setminus\{z_0\}$ and let k and j be two integers such that $k \neq j$ . Then $$m_{z_0}\left(r, \frac{f^{(k)}(z)}{f^{(j)}(z)}\right) \leq O\left(T_{z_0}(r, f) + \log\frac{1}{r}\right), \quad \text{for all } r \in (0, r_4] \setminus E_9 \quad \text{with } m_l(E_9) < \infty.$$

Lemma 11. ([21]). Assume $f \not\equiv 0$ is a solution of (2), set $g = f - \varphi$ . Then g satisfies <span id="page-8-4"></span> $$g^{(k)} + A_{k-1}g^{(k-1)} + \dots + A_1g' + A_0g = -\left[\varphi^{(k)} + A_{k-1}\varphi^{(k-1)} + \dots + A_1\varphi' + A_0\varphi\right]. \tag{15}$$

Lemma 12. ([14]). Let f be a non-constant meromorphic function in $\overline{\mathbb{C}} - \{z_0\}$ and set $g(\omega) = f(z_0 - \frac{1}{\omega})$ . Then $g(\omega)$ is meromorphic in $\mathbb{C}$ and we have $$T(R,g) = T_{z_0}(\frac{1}{R},f).$$

Lemma 13. ([1]). Let f be a meromorphic function in $\mathbb{C}$ with $p \geq q \geq 1$ . Then $$\sigma_{[p,q]}(f') = \sigma_{[p,q]}(f).$$

Lemma 14. Let f be a non-constant meromorphic function in $\overline{\mathbb{C}}\setminus\{z_0\}$ with $p\geq q\geq 1$ . Then $$\sigma_{[p,q]}(f^{(n)}, z_0) = \sigma_{[p,q]}(f, z_0), \qquad n \in \mathbb{N}.$$

Lemma 15. ([20]). Let f be a non-constant meromorphic function in $\overline{\mathbb{C}} - \{z_0\}$ . Then $$T_{z_0}(r, \frac{1}{f}) = T_{z_0}(r, f) + O(1).$$

Lemma 16. ([11]). Let $F(z) \not\equiv 0$ , $A_0(z), ..., A_{k-1}(z)$ be analytic functions in $\overline{\mathbb{C}} \setminus \{z_0\}$ and let f be a non-constant analytic solution in $\overline{\mathbb{C}} \setminus \{z_0\}$ of the equation $$f^{(k)} + A_{k-1}(z)f^{(k-1)} + \dots + A_1(z)f' + A_0(z)f = F(z), \tag{16}$$ such that $$\max \left\{ \sigma_{[2,2]}(F,z_0), \sigma_{[2,2]}(A_j,z_0) : (j=0,...,k-1) \right\} < \sigma_{[2,2]}(f,z_0).$$ Then λ[2,2](f, z0) = λ[2,2](f, z0) = σ[2,2](f, z0).