  We revisit and prove some convexity inequalities for trace functions
conjectured in the earlier part I. The main functional considered is
  \Phi_{p,q}(A_1,A_2,...,A_m) = (trace((\sum_{j=1}^m A_j^p)^{q/p}))^{1/q} for m
positive definite operators A_j. In part I we only considered the case q=1 and
proved the concavity of \Phi_{p,1} for 0 < p \leq 1 and the convexity for p=2.
We conjectured the convexity of \Phi_{p,1} for 1< p < 2. Here we not only
settle the unresolved case of joint convexity for 1 \leq p \leq 2, we are also
able to include the parameter q\geq 1 and still retain the convexity. Among
other things this leads to a definition of an L^q(L^p) norm for operators when
1 \leq p \leq 2 and a Minkowski inequality for operators on a tensor product of
three Hilbert spaces -- which leads to another proof of strong subadditivity of
entropy. We also prove convexity/concavity properties of some other, related
functionals.
