Weighted Differentiation Composition Operator from Logarithmic Bloch Spaces to Zygmund-Type Spaces

Abstract

Let $H(𝔻)$ denote the space of all holomorphic functions on the unit disk $𝔻$ of $ℂ$, $u\in H(𝔻)$ and let n be a positive integer, $\phi$ a holomorphic self-map of $𝔻$, and $\mu$ a weight. In this paper, we investigate the boundedness and compactness of a weighted differentiation composition operator ${𝒟}_{\phi ,u}^{n}f(z)=u(z)f^{(n)}(\phi (z)),f\in H(𝔻)$, from the logarithmic Bloch spaces to the Zygmund-type spaces.