A Minkowski Type Trace Inequality and Strong Subadditivity of Quantum Entropy

We consider the following trace function on n-tuples of positive operators: $${\Phi _P}({A_1},{A_2},...,{A_n}) = Tr({(\sum\limits_{j = 1}^n {A_j^P} )^{1/P}})$$ and prove that it is jointly concave for 0 2, Фp is neither convex nor concave. We conjecture that Фp is convex for 1 < p < 2, but our methods do not show this.