Weighted composition followed by differentiation operators on Zygmund spaces

Let $\mathcal{H}(\BD)$ denote the space of analytic functions on the open unit disk $\BD$. Let $u\in \mathcal{H}(\BD)$ and $\var$ be an analytic self-map of $\BD$. The product of the differentiation operator and the weighted composition operator $uC_\var$ on $\mathcal{H}(\BD)$ is defined by $$D uC_{\var}(f)=(u\cdot (f\circ \var)) =u (f\circ \var)+u\var (f \circ \var), \ f\in\mathcal{H}(\BD). $$ They are called weighted composition followed by differentiation operators. In this paper, we characterize the boundedness and compactness of $DuC_\var$ acting on Zygmund spaces, in terms of $u, \var$, their derivatives and the n-th power $\var^n$ of $\var.$