Lieb's simple proof of concavity of (A, B) → Tr Ap K†B1-pKand remarks on related inequalities

A simple, self-contained proof is presented for the concavity of the map (A, B) -> TrA(p)K(dagger)B(1-p)K. The author makes no claim to originality; this paper gives Lieb's original argument in its simplest, rather than its most general, form. A sketch of the chain of implications from this result to concavity of A -> Tr e(K+logA) is then presented. An independent elementary proof is given for the joint convexity of the map (A, B, X) -> Tr integral(infinity)(0) X-dagger (1)/X-A + uI(1)/(B + uI)du, which plays a key role in entropy inequalities.