  This paper is devoted to improvements of Sobolev and Onofri inequalities. The
additional terms involve the dual counterparts, i.e. Hardy-Littlewood-Sobolev
type inequalities. The Onofri inequality is achieved as a limit case of Sobolev
type inequalities. Then we focus our attention on the constants in our improved
Sobolev inequalities, that can be estimated by completion of the square
methods. Our estimates rely on nonlinear flows and spectral problems based on a
linearization around optimal Aubin-Talenti functions.
