The essential norm of a weighted composition operator on BMOA

We provide an estimate for the essential norm of a weighted composition operator $W_{\psi,\varphi}\colon f\mapsto \psi(f\circ\varphi)$ acting on the space $BMOA$ in terms of the weight function $\psi$ and the $n$-th power $\varphi^n$ of the analytic self-map $\varphi$ of the open unit disc $\mathbb{D}$. We also provide a new estimate for the norm of the weighted composition operator on $BMOA$.