Weighted composition operators on the class of subordinate functions

TL;DR

This paper investigates weighted composition operators that preserve a specific class of subordinate analytic functions, exploring their properties, examples, and fixed points to deepen understanding of their structure and behavior.

Contribution

It introduces new results on the preservation and fixed points of weighted composition operators acting on the class of subordinate functions.

Findings

Characterization of operators preserving the class al_{lpha}

Examples illustrating the preservation properties

Analysis of fixed points of these operators

Abstract

In this article, we study the weighted composition operators preserving the class $\mathcal{P}_{\alpha}$ of analytic functions subordinate to $\frac{1+\alpha z}{1-z}$ for $|\alpha|\leq 1, \alpha \neq -1$. Some of its consequences and examples for some special cases are presented. Furthermore, we discuss about the fixed points of weighted composition operators.