Eigenfunctions of Composition Operators on Bloch-type Spaces

Suppose $\phi$ is a holomorphic self map of the unit disk and $C_{\phi }$ is a composition operator with symbol $\phi$ that fixes the origin and $0<\mathrm{\mid }{\phi }^{\mathrm{\prime }}(0)\mathrm{\mid }<1$. This work explores sufficient conditions that ensure all holomorphic solutions of Schr"oder equation for the composition operator $C_{\phi }$ belong to a Bloch-type space ${\mathcal{B}}_{\alpha }$ for some $\alpha >0$. The results from composition operators have been extended to weighted composition operators in the second part of this work.