  Wehrl used Glauber coherent states to define a map from quantum density
matrices to classical phase space densities and conjectured that for Glauber
coherent states the mininimum classical entropy would occur for density
matrices equal to projectors onto coherent states. This was proved by Lieb in
1978 who also extended the conjecture to Bloch SU(2) spin-coherent states for
every angular momentum $J$. This conjecture is proved here. We also recall our
1991 extension of the Wehrl map to a quantum channel from $J$ to $K=J+1/2, J+1,
...$, with $K=\infty$ corresponding to the Wehrl map to classical densities.
For each $J$ and $J<K\leq \infty$ we show that the minimal output entropy for
these channels occurs for a $J$ coherent state. We also show that coherent
states both Glauber and Bloch minimize any concave functional, not just
entropy.
