In this talk, we extend the classical Schwarz-Pick estimate in several new directions, and establish some inequalities for holomorphic functions avoiding the origin. We first prove k-th order derivative estimates for holomorphic functions from the unit disk to the punctured disk. This result is then generalized to higher dimension complex space, where a Schwarz-Pick type inequality is obtained for nonvanshing holomorphic functions in the unit ball. As an application, we also give a Lojasiewicz-type inequality in C^n, providing a low bound for holomorphic functions near their zero sets.