Non-commutative Clarkson inequalities for unitarily invariant norms

It is shown that if A and B are operators on a separable complex Hilbert space and if \\\.\\\ is any unitarily invariant norm, then 2\\\ \A\(p) + \B\(p) \\\ less than or equal to \\\ \A + B\(p) + \A - B\(p) \\\ less than or equal to2(p-1) \\\ \A\(p) + \B\(p) \\\ for 2 less than or equal to p < ∞, and 2(p-1) ||| |A|(p) + |B|(p) ||| ≤ ||| |A + B|(p) + |A - B|(p) ||| ≤2 ||| |A|(p) + |B|(p) ||| for 0 < p less than or equal to 2. These inequalities are natural generalizations of some of the classical Clarkson inequalities for the Schatten p-norms. Generalizations of these inequalities to larger classes of functions including the power functions are also obtained.