Herman's theory revisited

We prove that a C (2+alpha) -smooth orientation-preserving circle diffeomorphism with rotation number in Diophantine class D (delta) , 0a parts per thousand currency sign delta <alpha a parts per thousand currency sign1, alpha-delta not equal 1, is C (1+alpha-delta) -smoothly conjugate to a rigid rotation. This is the first sharp result on the smoothness of the conjugacy. We also derive the most precise version of Denjoy's inequality for such diffeomorphisms.