A criterion for univalence extension

This paper presents a study on the univalence extension of a specific class of functions defined by an integral operator. The research provides sufficient conditions for determining this univalence extension, which depend on two arbitrary analytic functions defined in the unit disk, alongside three complex numbers. The main result of the study includes these functions and parameters, and by specializing them, it leads to various well-known univalent conditions. The paper showcases examples to demonstrate how these conditions can be applied.