Suppose $(M,\omega_{\varphi})$ is a constant scalar curvature Kähler metric. Then the following statements are mutually equivalent:

1. (1)

   There is a constant such that $\int_{M}\;\log{\omega_{\varphi}^{n}\over\omega^{n}}\cdot\omega_{\varphi}^{n}<C;$

2. (2)

   There is a constant such that $|\varphi|<C;$

3. (3)

   There is a constant $C$ such that $|\nabla\varphi|<C$ and $\log{\omega_{\varphi}^{n}\over\omega^{n}}\geq-C$;

4. (4)

   There is a constant $C$ such that $\frac{1}{C}<{\omega_{\varphi}^{n}\over\omega^{n}}<C;$

5. (5)

   There is a constant $C$ such that $n+\Delta\varphi<C$ and $\frac{\omega_{\varphi}^{n}}{\omega_{0}^{n}}>\frac{1}{C}$;

6. (6)

   All higher derivates of $\varphi$ is uniformly bounded.