Inequalities for L^p-norms that sharpen the triangle inequality and complement Hanner's Inequality

In 2006 Carbery raised a question about an improvement on the naïve norm inequality ∥f+g∥^p_p ≤ 2^(p−1)(∥f∥^p_p+∥g∥^p_p) for two functions in L^p of any measure space. When f=g this is an equality, but when the supports of f and g are disjoint the factor 2^(p−1) is not needed. Carbery's question concerns a proposed interpolation between the two situations for p>2. The interpolation parameter measuring the overlap is ∥fg∥_(p/2). We prove an inequality of this type that is stronger than the one Carbery proposed. Moreover, our stronger inequalities are valid for all p.