Abstract:

Let $H({\Bbb D})$ denote the space of all analytic functions on the unit disk ${\Bbb D}$ in the complex plane ${\Bbb C}$, $\psi_1, \psi_2\in H({\Bbb D}),$ $n$ be a nonnegative integer, $\varphi$ an analytic self-map of ${\Bbb D}$ and $\mu$ a weight. We study the boundedness and compactness of a product-type operator which is defined by

$T^n_{\psi_{1},\psi_{2},\varphi}f(z)=\psi_1(z)f^{(n)}(\varphi(z))+\psi_2(z)f^{(n+1)}(\varphi(z)), f\in H({\Bbb D}),$

from the mixed norm space to Zygmund-type spaces.

Key words: Product-type operator, Mixed norm space, Zygmund-type space, Cramer's Rule