  Using integral formulas based on Green's theorem and in particular a lemma of
Uchiyama, we give simple proofs of comparisons of different BMO norms without
using the John-Nirenberg inequality while we also give a simple proof of the
strong John-Nirenberg inequality. Along the way we prove the inclusions of BMOA
in the dual of H^1 and BMO in the dual of real H^1.
