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 1. Let A0(z), ..., A^k−1(z) be analytic functions in C\{z0} of finite logarithmic order with max σlog(A^j , z0) : j 6= 0 ≤ µlog(A0, z0) ≤ σlog(A0, z0) < +∞ and $$\sum_{\sigma_{\log}(A_j, z_0) = \mu_{\log}(A_0, z_0) \ge 1, j \ne 0} \tau_{\log}(A_j, z_0) < \underline{\tau}_{\log}(A_0, z_0) < +\infty.$$ Then, every analytic solution f(z)(6≡ 0) in C\{z0} of [\(2\)](#page-3-1) satisfies 0 ≤ µlog(A0, z0) − 1 ≤ µ[2,2](f, z0) ≤ µlog(A0, z0), and µlog(A0, z0) = µ[2,2](f, z0) ≤ σ[2,2](f, z0) = σlog(A0, z0) = λ[2,2](f − ϕ, z0) = λ[2,2](f − ϕ, z0) if µlog(A0, z0) > 1, where ϕ(z)(6≡ 0) is an analytic function in C\{z0} satisfying σ[2,2](ϕ, z0) < µlog(A0, z0).

Theorem 2. Let A0(z), ..., A^k−1(z) be analytic functions in C\{z0} of finite logarithmic order with max σlog(A^j , z0) : j 6= 0 ≤ µlog(A0, z0) ≤ σlog(A0, z0) < +∞ and $$\limsup_{r \longrightarrow 0} \frac{\sum_{j \neq 0} m_{z_0}(r, A_j)}{m_{z_0}(r, A_0)} < 1.$$ Then, every analytic solution f(z)(6≡ 0) in C\{z0} of [\(2\)](#page-3-1) satisfies 0 ≤ µlog(A0, z0) − 1 ≤ µ[2,2](f, z0) ≤ µlog(A0, z0), and µlog(A0, z0) = µ[2,2](f, z0) ≤ σ[2,2](f, z0) = σlog(A0, z0) = λ[2,2](f − ϕ, z0) = λ[2,2](f − ϕ, z0) if µlog(A0, z0) > 1, where ϕ(z)(6≡ 0) is an analytic function in C\{z0} satisfying σ[2,2](ϕ, z0) < µlog(A0, z0).

Theorem 3. Let A0(z), ..., A^k−1(z) be meromorphic functions in C\{z0} of finite logarithmic order with $$\liminf_{r \to 0} \frac{m_{z_0}(r, A_0)}{T_{z_0}(r, A_0)} = \delta > 0, \quad \max \left\{ \sigma_{\log}(A_j, z_0) : j \neq 0 \right\} \le \mu_{\log}(A_0, z_0) \le \sigma_{\log}(A_0, z_0) < +\infty$$ and $$\sum_{\sigma_{\log}(A_j, z_0) = \mu_{\log}(A_0, z_0) \ge 1, j \ne 0} \tau_{\log}(A_j, z_0) < \delta_{\underline{\tau}_{\log}}(A_0, z_0) < +\infty.$$ Then, every meromorphic solution f(z)(6≡ 0) in C\{z0} of [\(2\)](#page-3-1) satisfies 0 ≤ µlog(A0, z0) − 1 ≤ µ[2,2](f, z0), and µlog(A0, z0) ≤ µ[2,2](f, z0) if µlog(A0, z0) > 1.

Theorem 4. Let A0(z), ..., A^k−1(z) be meromorphic functions in C\{z0} of finite logarithmic order with $$\liminf_{r \longrightarrow 0} \frac{m_{z_0}(r,A_0)}{T_{z_0}(r,A_0)} = \delta > 0 \quad and \quad \limsup_{r \longrightarrow 0} \frac{\sum_{j \neq 0} m_{z_0}(r,A_j)}{m_{z_0}(r,A_0)} < 1.$$ Then, every meromorphic solution f(z)(6≡ 0) in C\{z0} of [\(2\)](#page-3-1) satisfies 0 ≤ µlog(A0, z0) − 1 ≤ µ[2,2](f, z0) and µlog(A0, z0) ≤ µ[2,2](f, z0) if µlog(A0, z0) > 1.

Theorem 5. Let A0(z), ..., A^k−1(z) be meromorphic functions in C\{z0} of finite logarithmic order with λlog( A⁰ , z0) + 1 < µlog(A0, z0), max σlog(A^j , z0) : j 6= 0 ≤ µlog(A0, z0) ≤ σlog(A0, z0) < +∞ and $$\sum_{\sigma_{\log}(A_j, z_0) = \mu_{\log}(A_0, z_0), j \neq 0} \tau_{\log}(A_j, z_0) < \underline{\tau}_{\log}(A_0, z_0) < +\infty.$$ Then, every meromorphic solution f(z)(6≡ 0) in C\{z0} of [\(2\)](#page-3-1) satisfies 1 < µlog(A0, z0) ≤ µ[2,2](f, z0).

Theorem 6. Let A0(z), ..., Ak−1(z) be analytic functions in C\{z0} of finite logarithmic order. Suppose there exists an integer s (0 ≤ s ≤ k − 1) such that As(z) satisfies max σlog(A^j , z0) : j 6= s ≤ µlog(As, z0) < +∞ and $$\sum_{\sigma_{\log}(A_j, z_0) = \mu_{\log}(A_s, z_0) \ge 1, j \ne s} \tau_{\log}(A_j, z_0) < \underline{\tau}_{\log}(A_s, z_0) < +\infty.$$ Then, every meromorphic solution $f(z) (\not\equiv 0)$ in $\overline{\mathbb{C}} \setminus \{z_0\}$ of (2) satisfies $\mu_{[2,2]}(f,z_0) - 1 \leq \mu_{\log}(A_s,z_0) - 1 \leq \mu_{\log}(f,z_0)$ . Further, if $\mu_{\log}(A_s,z_0) > 1$ , then $\mu_{[2,2]}(f,z_0) \leq \mu_{\log}(A_s,z_0) \leq \mu_{\log}(f,z_0)$ . Remark 2. As for the case when the dominant coefficient is $A_0(z)$ , we can replace the condition $$\sum_{\sigma_{\log}(A_j, z_0) = \mu_{\log}(A_s, z_0) \ge 1, j \ne s} \tau_{\log}(A_j, z_0) < \underline{\tau}_{\log}(A_s, z_0) < +\infty$$ $\label{eq:continuous_problem} \textit{in Theorem 6 by } \limsup_{r\longrightarrow 0} \frac{\sum_{j\neq s} m_{z_0}(r, A_j)}{m_{z_0}(r, A_s)} < 1.$

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 $$\sigma_{[p,q]}(f,z_0) = \limsup_{r \to 0} \frac{\log_p^+ V_{z_0}(r,f)}{\log_q \frac{1}{r}}, \qquad \mu_{[p,q]}(f,z_0) = \liminf_{r \to 0} \frac{\log_p^+ V_{z_0}(r,f)}{\log_q \frac{1}{r}}.$$

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 $$\frac{f^{(j)}(z_r)}{f(z_r)} = \left(\frac{V_{z_0}(r,f)}{z_0 - z_r}\right)^j (1 + o(1)),$$ 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 $$\lim_{r \to 0} \frac{\log \log M_{z_0}(r, f)}{\log \log \frac{1}{r}} = \lim_{r \to 0} \frac{\log T_{z_0}(r, f)}{\log \log \frac{1}{r}} = \sigma.$$ (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 $$\left| \frac{f^{(j)}(z)}{f(z)} \right| \le C \left[ \frac{1}{r^2} T_{z_0}(\lambda r, f) \log T_{z_0}(\lambda r, f) \right]^j. \tag{8}$$ (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 $$g^{(k)} + A_{k-1}g^{(k-1)} + \dots + A_1g' + A_0g = -\left[\varphi^{(k)} + A_{k-1}\varphi^{(k-1)} + \dots + A_1\varphi' + A_0\varphi\right]. \tag{15}$$

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).

Definition 4. Let f be a meromorphic function in C − {z0}, the logarithmic lower order of f near z⁰ is given by $$\mu_{[1,2]}(f,z_0) = \mu_{\log}(f,z_0) = \liminf_{r \to 0} \frac{\log^+ T_{z_0}(r,f)}{\log \log \frac{1}{r}},$$ and if µlog(f, z0) ∈ [1, +∞), then the logarithmic lower type of f near z⁰ is given by $$\underline{\tau}_{[1,2]}(f,z_0) = \underline{\tau}_{\log}(f,z_0) = \liminf_{r \to 0} \frac{T_{z_0}(r,f)}{(\log \frac{1}{r})^{\mu_{\log}(f,z_0)}}.$$ If f is an analytic function in C − {z0}, then $$\mu_{[1,2]}(f,z_0) = \mu_{\log}(f,z_0) = \liminf_{r \to 0} \frac{\log^+ \log^+ M_{z_0}(r,f)}{\log \log \frac{1}{r}},$$ and if µlog(f, z0) ∈ [1, +∞), then the logarithmic lower type of f near z⁰ is given by $$\underline{\tau}_{[1,2],M}(f,z_0) = \underline{\tau}_{\log,M}(f,z_0) = \liminf_{r \to 0} \frac{\log^+ M_{z_0}(r,f)}{(\log \frac{1}{r})^{\mu_{\log}(f,z_0)}}.$$ Remark 1. According to (Lemma 2.2, [\[14\]](#page-16-0)) if f is a non constant meromorphic function in C − {z0}, then g(ω) = f(z⁰ − ¹ ω ) is meromorphic in C and they satisfy $$T(R,g) = T_{z_0}(\frac{1}{R},f).$$ Consequently all the properties of the logarithmic order for the meromorphic functions in C are hold, such as all the non-constant rational functions which are analytic in C − {z0} are of logarithmic order equals one, where there is no transcendental meromorphic function in C− {z0} of logarithmic order less than one, further, constant functions have zero logarithmic order and there are no meromorphic functions in C − {z0} of logarithmic order between zero and one (see, [\[8,](#page-16-10) [9\]](#page-16-11)).

Definition 5. The [p, q] exponent of convergence of the sequence of zeros and distinct zeros of a meromorphic function f in C − {z0} are respectively defined by $$\lambda_{[p,q]}(f,z_0) = \limsup_{r \to 0} \frac{\log_p^+ N_{z_0}(r,\frac{1}{f})}{\log_q \frac{1}{r}}, \qquad \overline{\lambda}_{[p,q]}(f,z_0) = \limsup_{r \to 0} \frac{\log_p^+ \overline{N}_{z_0}(r,\frac{1}{f})}{\log_q \frac{1}{r}}.$$

Definition 6. The logarithmic exponent of convergence of the sequence of zeros and distinct zeros of a meromorphic function f in C − {z0} are respectively defined by $$\lambda_{\log}(f, z_0) = \limsup_{r \to 0} \frac{\log^+ N_{z_0}(r, \frac{1}{f})}{\log \log \frac{1}{r}} - 1, \qquad \overline{\lambda}_{\log}(f, z_0) = \limsup_{r \to 0} \frac{\log^+ \overline{N}_{z_0}(r, \frac{1}{f})}{\log \log \frac{1}{r}} - 1.$$ Before we start stating our results, we may mention some of the previous results obtained on the growth of solutions of linear differential equations in the extended complex plane except a finite singular point C\{z0} and we focus particularly on those involving the lower order. In [\[19\]](#page-16-8), Liu and his co-authors considered the following complex homogeneous second order linear differential equation $$f'' + A(z)f' + B(z)f = 0, (1)$$ where A(z) and B(z) are both analytic functions in C \ {z0}, and obtained the following theorems. Theorem A ([\[19\]](#page-16-8)). Let A(z) and B(z) be analytic functions in C\{z0} satisfying µ(A, z0) < µ(B, z0) < ∞. Then, every non trivial solution f(z) of [\(1\)](#page-3-0) that is analytic in C\{z0}, satisfies σ2(f, z0) ≥ µ(B, z0). Theorem B ([\[19\]](#page-16-8)). Let A(z) and B(z) be analytic functions in C\{z0} satisfying µ(A, z0) = µ(B, z0) with τ^M (A, z0) < τ^M (B, z0). Then, every non trivial solution f(z) of [\(1\)](#page-3-0) that is analytic in C\{z0}, satisfies σ2(f, z0) ≥ µ(B, z0). For more general case, in a recent paper [\[10\]](#page-16-3), we considered the following higher order linear differential equation $$f^{(k)} + A_{k-1}(z)f^{(k-1)} + \dots + A_1(z)f' + A_0(z)f = 0,$$ (2) where A^j (z) (j = 0, 1, . . . , k − 1) are analytic functions C − {z0} , and we obtained the following results. Theorem C ([\[10](#page-16-3)]). Let A0(z), ..., A^k−1(z) be analytic functions in C\{z0}. Assume that $$\max \left\{ \sigma_{[p,q]}(A_j, z_0) : j \neq 0 \right\} < \mu_{[p,q]}(A_0, z_0) \le \sigma_{[p,q]}(A_0, z_0) < +\infty.$$ Then, every analytic solution f(z)(6≡ 0) in C\{z0} of [\(2\)](#page-3-1) satisfies $$\mu_{[p,q]}(A_0, z_0) = \mu_{[p+1,q]}(f, z_0) \le \sigma_{[p+1,q]}(f, z_0) = \sigma_{[p,q]}(A_0, z_0).$$ Theorem D ([\[10](#page-16-3)]). Let A0(z), ..., A^k−1(z) be analytic functions in C\{z0}. Assume that $$\max \left\{ \sigma_{[p,q]}(A_j, z_0) : j \neq 0 \right\} \le \mu_{[p,q]}(A_0, z_0) \le \sigma_{[p,q]}(A_0, z_0) < +\infty$$ and $$\max\left\{\tau_{[p,q],M}(A_j,z_0):\sigma_{[p,q]}(A_j,z_0)=\mu_{[p,q]}(A_0,z_0)>0\right\}<\underline{\tau}_{[p,q],M}(A_0,z_0)<+\infty.$$ Then, every analytic solution f(z)(6≡ 0) in C\{z0} of [\(2\)](#page-3-1) satisfies µ[p,q](A0, z0) = µ[p+1,q](f, z0) ≤ σ[p+1,q](f, z0) = σ[p,q](A0, z0). As we mentioned before, in this article you may find several similarities between the results obtained in [\[13\]](#page-16-13) for complex plane C and the present results which we also consider them as extensions to the above theorems and other results to the case that all the coefficients are zero order analytic or meromorphic functions in C\{z0}. The first five theorems are for the special case that the coefficients of [\(2\)](#page-3-1) are analytic or meromorphic in C\{z0} and A0(z) dominates them, the rest of the results are for the case when the dominant coefficient is an arbitrary As(z). In fact, we obtain the following results.