An integral type operator on analytic function spaces

Abstract

Let f and h be analytic functions on the unit disk $\mathbb {D}$. For any positive integer n we define an integral type operator $I_{n, f}$ as

$$\begin{aligned} I_{n, f}(h)(z)=\int _{0}^{z}f(w)h^{(n)}(w)dw, \end{aligned}$$

where $f^{(n)}$ denote $n^{th}$ order derivative of f. In this article we discuss the boundedness of $I_{n, f}$ in the Möbius invariant spaces $Q_{p}$ for $0<p<1$ and find the norm of $I_{n,f}$ computed on $\alpha$-Bloch space ${\mathscr {B}}^{\alpha }$ for $\alpha >0$.