Abstract: In this article, we first establish an asymptotically sharp result on the higher order Fréchet derivatives for bounded holomorphic mappings $f(x)=f(0)+\sum\limits_{s=1}^\infty\frac{D^{sk} f(0)(x^{sk})}{(sk) !}: B_X\rightarrow B_Y$, where $B_X$ is the unit ball of $X$. We next give a sharp result on the first order Fréchet derivative for bounded holomorphic mappings $f(x)=f(0)+\sum\limits_{s=k}^\infty\frac{D^{s} f(0)(x^{s})}{s !}: B_X\rightarrow B_Y$, where $B_X$ is the unit ball of $X$. The results that we derive include some results in several complex variables, and extend the classical result in one complex variable to several complex variables.

Key words: refined Schwarz-Pick estimate, bounded holomorphic mapping, Carathéodory metric, first order Fréchet derivative, higher order Fréchet derivatives