For $\sum s_{i}^{2}\lesssim 1$,

$$\begin{cases}|I_{\Gamma,f}(s_{1},s_{2},t_{1},t_{2},t_{3})+\frac{f(s_{1},s_{2})}{4\pi R}|\leq C,\\ |\frac{\partial I_{\Gamma,f}}{\partial t_{i}}-\frac{f(s_{1},s_{2})t_{i}}{4\pi R^{3}}(1+\frac{1}{5}(\sum_{j}t_{j}\vec{n}_{j})\cdot\vec{H})|\leq C,\quad&i=3,4,5,\\ |\frac{\partial I_{\Gamma,f}}{\partial s_{i}}+\frac{1}{4\pi R}\frac{\partial f}{\partial s_{i}}(s_{1},s_{2})|\leq C,\quad&i=1,2.\end{cases}$$

where $\vec{H}$ is the mean curvature vector of $\Gamma$ at $\vec{h}(s_{1},s_{2})$. If morever $\left\lVert f\right\rVert_{C^{3}}\leq C$, $\left\lVert f_{i}\right\rVert_{C^{3}}\leq C$, then

$$|\frac{\partial^{2}I_{\Gamma,f}}{\partial t_{i}\partial t_{j}}-\frac{(\delta_{ij}R^{2}-3t_{i}t_{j})f}{4\pi R^{5}}|\leq\frac{C}{R^{2}},\quad|\frac{\partial^{2}I_{\Gamma,f}}{\partial t_{i}\partial s_{j}}-\frac{t_{i}}{4\pi R^{3}}\frac{\partial f}{\partial s_{j}}|\leq\frac{C}{R},\quad|\frac{\partial^{2}I_{\Gamma,f}}{\partial s_{i}\partial s_{j}}|\leq\frac{C}{R}.$$

If morever $f(0)=0$, then at $s_{1}=s_{2}=0$,

$$|\frac{\partial^{2}I_{\Gamma,f}}{\partial t_{i}\partial t_{j}}|\leq\frac{C}{R},\quad|\frac{\partial^{2}I_{\Gamma,f}}{\partial s_{i}\partial s_{j}}+\frac{1}{4\pi R}\frac{\partial^{2}f}{\partial s_{i}\partial s_{j}}|\leq C.$$

If morever $df(0)=0$, then $|\frac{\partial^{2}I_{\Gamma,f}}{\partial t_{i}\partial s_{j}}|\leq C$ for $s_{1}=s_{2}=0$.