Einstein four-manifolds of positive sectional curvature and ADM mass

We prove that closed Einstein four-manifolds $(M,g)$ with positive sectional curvature and Euler characteristic and signature satisfying $2\chi (M)-3\mathrm{\mid }\tau (M)\mathrm{\mid }\le 4$ must be homothetically isometric to the round $S^4$ or to ${\mathbb{C}\mathbb{P}}^2$ with the Fubini-Study metric. The proof relies on an upper bound on the ADM mass of conformal blow-ups of $(M,g)$, combined with a lower one from \cite{GM}.