We are interested in sharp functional inequalities for the coherent state
transform related to the Wehrl conjecture and its generalizations. This
conjecture was settled by Lieb in the case of the Heisenberg group and then by
Lieb and Solovej for SU(2) and by Kulikov for SU(1,1) and the affine group. In
this paper, we give alternative proofs and characterize, for the first time,
the optimizers in the general case. We also extend the recent Faber–Krahn-type
inequality for Heisenberg coherent states, due to Nicola and Tilli, to the
SU(2) and SU(1,1) cases. Finally, we prove a family of reverse H\”older
inequalities for polynomials, conjectured by Bodmann.

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