Following Zeriahi [40] one can prove that for all $\alpha<2/\nu(X,\omega)$ there exists $C_{\alpha}>0$ such that

$$\text{Vol}_{\omega}(\cdot)\leq C_{\alpha}T_{\omega}(\cdot)^{\alpha},$$

where $\nu(X,\omega)=\sup\{\nu(\varphi,x)\,/\,\varphi\in PSH(X,\omega),x\in X\}$ and $\nu(\varphi,x)$ denotes the Lelong number of $\varphi$ at point $x$. In particular it follows from proposition 3.8 that $\forall\varphi\in PSH(X,\omega)$ with $\sup_{X}\varphi=0$,

$$\text{Vol}_{\omega}(\varphi<-t)\leq C_{\alpha}\exp(-\alpha t),\;\forall t\in\mathbb{R}.$$

Such inequalities are quite useful in complex dynamics [20],[22] and in the study of the complex Monge-Ampère operator [24], [28].