New product type operators acting between weighted Bergman–Orlicz spaces and some weighted-type spaces

Consider a complex plane \({\mathbb {C}}\) $\mathbb{C}$ and open unit disk \({\mathbb {D}}= \{z\in {\mathbb {C}}:|z|<1\}\) $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$ . Let \(\phi _{1}\) ${\varphi }_1$ , \(\phi _{2}\) ${\varphi }_2$ be two holomorphic functions on \({\mathbb {D}}\) $\mathbb{D}$ , and let \(\xi\) $\xi$ be a holomorphic self-map of \({\mathbb {D}}\) $\mathbb{D}$ . For \(n\in {\mathbb {N}}_{0}\) $n\in {\mathbb{N}}_0$ , where \({\mathbb {N}}_{0}=\{0,1,2,\dots \}\) ${\mathbb{N}}_0=\{0,1,2,\cdots \}$ , a new product type operator on \(H({\mathbb {D}})\) $H(\mathbb{D})$ is defined by: \(\begin{aligned} T_{\phi _{1},\phi _{2},\xi }^{n}f(z)= \phi _{1}(z)f^{(n)}(\xi (z))+ \phi _{2}(z)f^{(n+1)}(\xi (z)), \;\;\; f\in H({\mathbb {D}}), \;z\in {\mathbb {D}}, \end{aligned}\) $$\begin{array}{c}\hfill T_{{\varphi }_1,{\varphi }_2,\xi }^{n}f(z)={\varphi }_1(z)f^{(n)}(\xi (z))+{\varphi }_2(z)f^{(n+1)}(\xi (z)),f\in H(\mathbb{D}),z\in \mathbb{D},\end{array}$$ where \(f^{(k)}\) $f^{(k)}$ denotes the kth derivative of the function f. In this paper, we investigate the boundedness and compactness of these type of operators acting between weighted Bergman–Orlicz spaces and some weighted-type spaces.