Let $\varphi$ be a plurisubharmonic function such that $\sup\varphi=-1$. Then

$$\int_{X}{\rm exp}\Big\{c(n,p,\gamma)\alpha\Big(\frac{-\varphi}{E_{p}(\varphi)^{\frac{1}{n+p}}}\Big)^{\frac{n+p}{n}}\Big\}\omega_{X}^{n}\leq 2C_{\alpha}$$

(6.2)

where $\alpha$ and $C_{\alpha}$ are the constants coming from the $\alpha$-invariant estimate of $(X,\omega_{X})$.