  The Wehrl entropy conjecture for coherent (highest weight) states in
representations of the Heisenberg group, which was proved in 1978 and recently
extended by us to the group $SU(2)$, is further extended here to symmetric
representations of the groups $SU(N)$ for all $N$. This result gives further
evidence for our conjecture that highest weight states minimize group integrals
of certain concave functions for a large class of Lie groups and their
representations.
