On weighted compositions preserving the Carath\'eodory class

TL;DR

This paper characterizes weighted composition transformations that preserve the class of normalized analytic functions with positive real part, providing criteria, analyzing special cases, and identifying fixed points.

Contribution

It offers a comprehensive characterization of such transformations, including criteria for preservation, analysis of special cases, and fixed point identification.

Findings

Criteria for weighted composition transformations preserving ${\\mathcal P}$

Identification of fixed points of these transformations

Analysis of special cases of symbols

Abstract

We characterize in various ways the weighted composition transformations which preserve the class ${\mathcal P}$ of normalized analytic functions in the disk with positive real part. We analyze the meaning of the criteria obtained for various special cases of symbols and identify the fixed points of such transformations.