  Friedland (1981) showed that for a nonnegative square matrix A, the spectral
radius r(e^D A) is a log-convex functional over the real diagonal matrices D.
He showed that for fully indecomposable A, log r(e^D A) is strictly convex over
D_1, D_2 if and only if D_1-D_2 != c I for any c \in R. Here the condition of
full indecomposability is shown to be replaceable by the weaker condition that
A and A'A be irreducible, which is the sharpest possible replacement condition.
Irreducibility of both A and A'A is shown to be equivalent to irreducibility of
A^2 and A'A, which is the condition for a number of strict inequalities on the
spectral radius found in Cohen, Friedland, Kato, and Kelly (1982). Such
`two-fold irreducibility' is equivalent to joint irreducibility of A, A^2, A'A,
and AA', or in combinatorial terms, equivalent to the directed graph of A being
strongly connected and the simple bipartite graph of A being connected.
Additional ancillary results are presented.
