  We show that certain monotone functionals on the Hardy spaces and convex
functionals on the Bergman spaces are maximized at the normalized reproducing
kernels among the functions of norm $1$, thus proving the contractivity
conjecture of Pavlovi\'c and of Brevig, Ortega-Cerd\`a, Seip and Zhao and the
Wehrl-type entropy conjecture for the $SU(1,1)$ group of Lieb and Solovej,
respectively.
