  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.
