Non-commutative Clarkson inequalities for n-tuples of operators

Let A0,..., A(n-1) be operators on a separable complex Hilbert space H, and let alpha(0),..., alpha(n-1) be positive real numbers such that Sigma (n-1)(j=0) alpha= 1. We prove that for every unitarily invariant norm, [GRAPHICS] for 2 <= p < infinity, and the reverse inequality holds for 0 < p <= 2. Moreover, we prove that if omega(0),...,omega(n-1) are the n roots of unity with omega(j) = e(2 pi ij/n), 0 <= j <= n - 1, then for every unitarily invariant norm, [GRAPHICS] for 2 <= p < infinity, and the reverse inequalities hold for 0 < p <= 2. These inequalities, which involve n-tuples of operators, lead to natural generalizations and refinements of some of the classical Clarkson inequalities in the Schatten p-norms. Extensions of these inequalities to certain convex and concave functions, including the power functions, are olso optained.