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

A new family of sharp conformally invariant integral inequalities