Generalized Weighted Composition Operators From Logarithmic Bloch Type Spaces to $ n $'th Weighted Type Spaces

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 $fin mathcal{H}(mathbb{D}) $ such that $sup_{zin mathbb{D}}mu(z)left|f^{(n)}(z)right|begin{align*}left|f right|_{mathcal{W}_mu ^{(n)}}=sum_{j=0}^{n-1}left|f^{(j)}(0)right|+sup_{zin 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 $varphi(mathbb{D})subseteqmathbb{D}$. We also provide an estimation for the essential norm of these operators.