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$isameromorphicfunctioninaregionoftheform$ 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.