Estimates of the logarithmic derivative near a singular point and applications

In this paper, we will give estimates for the logarithmic derivative $\left\vert \frac{f^{\left( k\right) }\left( z\right) }{f\left( z\right) } \right\vert$ where $f$ is a meromorphic function in a region of the form $D\left( 0,R\right) =\left\{ z\in \mathbb{C} :0<\left\vert z\right\vert <R\right\} .$ Some applications on the growth of solutions of linear differential equations near a singular point are given.