Estimates of essential norms of weighted composition operator from Bloch type spaces to Zygmund type spaces

Let $u$ be a holomorphic function and $\phi$ a holomorphic self-map of the open unit disk $\mathbb{D}$ in the complex plane. We give some new characterizations for the boundedness of the weighted composition operators $u{C}_{\phi }$ from Bloch type spaces to Zygmund type spaces in $\mathbb{D}$ in terms of $u,\phi$, their derivatives and the $n$-th power ${\phi }^n$ of $\phi$. Moreover, we obtain some similar estimates for their essential norms. From which the sufficient and necessary conditions of compactness of the operators $u{C}_{\phi }$ follows immediately.