On the cyclic elements of the shift operator in a weighted anisotropic space of holomorphic function in the polydisc

Let $\varphi(r)=(\varphi_1(r_1),\dots,\varphi_n(r_n))$ be a vector-valued function on $\mathbf R^n_+$. A necessary and sufficiently condition is obtained for every $f\in H^\infty(\mathbf D^n)$, $f(z)\ne 0$, $z\in \mathbf D^n$ to be cyclic in the corresponding $L^p(\varphi)$ weighted space, where $\mathbf D^n$ is unit polydisc in $\mathbf C^n$.