Let $\varphi$ be a plurisubharmonic function in the unit ball $B_1\subset \mathbb{C}^n$ with $\varphi\le 0$. Suppose that the Lelong number $\nu(\varphi,0)<k$ for some $k>0$. Does it follow that there exists $\alpha>0$, possibly depending on $k$, such that $\int_{B_{\frac{1}{2}}}e^{-\alpha\varphi}dvol(z)\le C$?

Note that this does not immediately follow from the definition of Lelong number. Since $\nu(\varphi,0)=\lim\inf_{z\rightarrow0}\frac{\varphi(z)}{\log|z|}$, so $\nu(\varphi,0)<k$ apriori only implies $\varphi(z)-k\log|z|$ bounded from below along some sequence $z_i\rightarrow0$.

On the other hand, if we know $\nu(\varphi,0)=0$, then above is indeed true, since under this assumption, we would have $\varphi(0)$ bounded from below, and the result above follows from a Lemma of Hormander(Lemma 4.4 in his book on several complex variables).