  We refine Theorem A due to Gursky \cite{G3}. As applications, we give some
rigidity theorems on four-manifolds with postive Yamabe constant. In
particular, these rigidity theorems are sharp for our conditions have the
additional properties of being sharp. By this we mean that we can precisely
characterize the case of equality.
  We prove some classification theorems of four manifolds according to some
conformal invariants (see Theorems 1.3 and 1.6), which generalize the
conformally invariant sphere theorem of Chang-Gursky-Yang \cite{CGY}.
