We revisit and prove some convexity inequalities for trace functions conjectured in this paper’s antecedent. The main functional considered is
$ \Phi_{p,q} (A_1,\, A_2, \ldots, A_m) = \left({\rm Tr}\left\left( \, {\sum\limits_{j=1}^m A_j^p } \, \right) ^{q/p} \right] \right)^{1/q} $
for
m positive definite operators
A j . In our earlier paper, we only considered the case
q = 1 and proved the concavity of Φ
p,1 for 0 <
p ≤ 1 and the convexity for
p = 2. We conjectured the convexity of Φ
p,1 for 1 <
p < 2. Here we not only settle the unresolved case of joint convexity for 1 ≤
p ≤ 2, we are also able to include the parameter
q ≥ 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 ≤
p ≤ 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.