Product type operators acting between weighted Bergman spaces and Bloch type spaces

For analytic functions u, ψ in the unit disk ⅅ in the complex plane and an analytic self-map φ of ⅅ, we describe in this paper the boundedness and compactness of product type operators

\(T_{u,\psi,\varphi}f(z)=u(z)f(\varphi(z))+\psi(z)f^{\prime}(\varphi(z)), \quad z\in \mathbb{D},\) $$T_{u,\psi ,\phi }f(z)=u(z)f(\phi (z))+\psi (z)f^{\mathrm{\prime }}(\phi (z)),z\in \mathbb{D},$$

acting between weighted Bergman spaces induced by a doubling weight and a Bloch type space with a radial weight.