A new family of sharp conformally invariant integral inequalities

We prove a one-parameter family of sharp integral inequalities for functions on the $n$-dimensional unit ball. The inequalities are conformally invariant, and the sharp constants are attained for functions that are equivalent to a constant function under conformal transformations. As a limiting case, we obtain an inequality that generalizes Carleman’s inequality for harmonic functions in the plane to poly-harmonic functions in higher dimensions.