Document Type : Research Paper

Author

Faculty of Engineering, Ardakan University, P.O. Box 184, Ardakan, Iran.

Abstract

Let $\mathcal{H}(\mathbb{D})$ denote the space of analytic functions on the open unit disc $\mathbb{D}$. For a weight $\mu$ and a nonnegative integer $n$, the $n$'th weighted type space $\mathcal{W}_\mu ^{(n)}$ is the space of all $f\in \mathcal{H}(\mathbb{D})$ such that $\sup_{z\in \mathbb{D}}\mu(z)\left|f^{(n)}(z)\right|<\infty.$ Endowed  with the norm
\begin{align*}
\left\|f \right\|_{\mathcal{W}_\mu ^{(n)}}=\sum_{j=0}^{n-1}\left|f^{(j)}(0)\right|+\sup_{z\in \mathbb{D}}\mu(z)\left|f^{(n)}(z)\right|,
\end{align*}
the $n$'th weighted type space is a Banach space.  In this paper, we characterize the boundedness of  generalized weighted composition operators $\mathcal{D}_{\varphi ,u}^m$  from logarithmic Bloch type spaces $\mathcal{B}_{{{\log }^\beta }}^\alpha$ to $n$'th weighted type spaces $\mathcal{W}_\mu ^{(n)}$, where $u$ and $\varphi$ are analytic functions on  $\mathbb{D}$ and $\varphi(\mathbb{D})\subseteq\mathbb{D}$. We also provide an estimation for the essential norm of these operators.

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