On The Relationship Between The Logarithmic Lower Order of Coefficients and The Growth of Solutions of Complex Linear Differential Equations in C̄₀\

In this article, we study the growth of solutions of the homogeneous complex linear differential equation {equation*} f⁽ᵏ⁾+Aₖ₋₁(z)f⁽ᵏ⁻¹⁾+⋯+A₁(z)f^{}+ A₀(z)f=0, {equation*}% where the coefficients Aⱼ(z) (j=0,1,… ,k-1) are analytic or meromorphic functions in C̄₀\. 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.