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

Abdelkader Dahmani, Benharrat Belaïdi

Abstract

In this article, we study the growth of solutions of the homogeneous complex linear differential equation \begin{equation*} f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots+A_{1}(z)f^{\prime}+ A_{0}(z)f=0, \end{equation*}% where the coefficients $A_{j}(z)$ $(j=0,1,\ldots ,k-1)$ are analytic or meromorphic functions in $\overline{\mathbb{C}}\setminus\{z_{0}\}$. Under the sufficient condition that there exists one dominant coefficient by its logarithmic lower order or by its logarithmic lower type. We extend some precedent results due to Liu, Long and Zeng and others.

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

Abstract

In this article, we study the growth of solutions of the homogeneous complex linear differential equation \begin{equation*} f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots+A_{1}(z)f^{\prime}+ A_{0}(z)f=0, \end{equation*}% where the coefficients are analytic or meromorphic functions in . Under the sufficient condition that there exists one dominant coefficient by its logarithmic lower order or by its logarithmic lower type. We extend some precedent results due to Liu, Long and Zeng and others.
Paper Structure (1 section, 27 theorems, 109 equations)

This paper contains 1 section, 27 theorems, 109 equations.

Table of Contents

  1. Introduction and Main Results

Key Result

Theorem A

Let $A(z)$ and $B(z)$ be analytic functions in $\overline{\mathbb{C}}\backslash \{z_0\}$ satisfying $\mu(A, z_0)<\mu(B, z_0)<\infty.$ Then, every non trivial solution $f(z)$ of (equation 1.1) that is analytic in $\overline{\mathbb{C}}\backslash \{z_0\},$ satisfies $\sigma_{2}(f, z_0)\geq\mu(B, z_0).

Theorems & Definitions (47)

  • Definition 1: L3
  • Definition 2: D1
  • Definition 3: D2
  • Definition 4
  • Remark 1
  • Definition 5
  • Definition 6
  • Theorem A: L2
  • Theorem B: L2
  • Theorem C: D1
  • ...and 37 more