Wehrl inequalities for matrix coefficients of holomorphic discrete series

We prove Wehrl-type $L^2(G)-L^p(G)$ inequalities for matrix coefficients of vector-valued holomorphic discrete series of $G$, for even integers $p=2n$. The optimal constant is expressed in terms of Harish-Chandra formal degrees for the discrete series. We prove the maximizers are precisely the reproducing kernels.