On the norm of the weighted Berezin transform

We consider a weighted Berezin transform: $$B_{\alpha }:L^{\mathrm{\infty }}({\mathbb{B}}^n)\to \mathcal{B},\alpha >-1,$$ defined, for $f \in L^{\infty} \left( $\mathbb{B}$^n \right)$and$z \in $\mathbb{B}$^n$,by$$(B_\alpha f) (z) = c_\alpha \int_{$\mathbb{B}$^n} \frac{\left( 1-|z|^2 \right)^{n+1}}{|1 - \langle z, w \rangle|^{2n+2}} f(w) \left( 1-|w|^2 \right)^\alpha \ d v(w),$$where$c_{\alpha} = \frac{\Gamma(\alpha+n+1)}{\Gamma(\alpha+1)\pi^n}$,$v$ is the Lebesque measure and $\mathcal{B}$ is a Bloch-type space. We prove that $B_{\alpha }$ is bounded iff $\alpha >0$ and give the exact semi-norm of $B_{\alpha }$ for $0\leq\alpha \leq 2n+3.$