The $\alpha$-Invariant on CP^2#2\bar{CP^2}

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.