On sum of weighted differentiation composition operators from Bergman spaces with admissible weights to Zygmund type spaces

Let \({\mathbb D}\) $\mathbb{D}$ be the open unit disk in the complex plane. We characterize the boundedness and compactness of the sum of weighted differentiation composition operators \(\begin{aligned} (T_{\overrightarrow{\psi }, \varphi } f)(z)=\sum _{j=0}^{n}(D^j_{\psi _j, \varphi }f)(z)=\sum _{j=0}^n\psi _{j}(z) f^{(j)} (\varphi (z)),\quad z\in {\mathbb D}, \end{aligned}\) $$\begin{array}{c}\hfill (T_{\overrightarrow{\psi },\phi }f)(z)=\sum _{j=0}^n(D_{{\psi }_j,\phi }^{j}f)(z)=\sum _{j=0}^n{\psi }_j(z)f^{(j)}(\phi (z)),z\in \mathbb{D},\end{array}$$ where \(n\in {\mathbb N}_0\) $n\in {\mathbb{N}}_0$ , \(\psi _j\) ${\psi }_j$ , \(j\in \overline{0,n}\) $j\in \overline{0,n}$ , are holomorphic functions on \({\mathbb D}\) $\mathbb{D}$ , and \(\varphi \) $\phi$ , a holomorphic self-maps of \({\mathbb D}\) $\mathbb{D}$ , acting from Bergman spaces with admissible weights to Zygmund type spaces.