  In this paper, we apply the Tian-Yau-Zelditch expansion of the Bergman kernel
on polarized K\"ahler metrics to approximate plurisubharmonic functions and
compute the $\alpha$-invariant of $CP^2#2\bar{CP^2}$, which is exactly 1/3. In
addition we prove Tian's conjecture on the generalized Moser-Trudinger
inequality in a special case prescribed by the $\alpha$-invariant.
