An analysis of convexity and starlikeness attributes for Breaz integro-differential operator

The Geometric Theory of Analytic Functions (GTAF) is the attractive part of complex analysis, which correlates with the rest of the themes in mathematics. Its essential purpose is to formulate numerous classes of geometric analytic functions and to discuss their geometric attributes. In continuation, the association between operator theory and the GTAF area started to take shape and has remained a topic of wide attention today. In the previous century, operator theory was extended to the complex open unit disk and has been applied to propose diverse sorts of generalizations of normalized analytic functions. As a result, the operator theory appears to be a good way to look for things in the GTAF area. Since then, the acquisition of geometric attributes by employing operators has become a significant theme of research studies. The current study centers on and investigates, in the classes of $\ell$-uniformly convex and starlike functions of order $\beta$, the convexity attribute by utilizing a modified Breaz integro-differential operator in the unit disk. Furthermore, in the class of analytic functions, some conditions that make the Breaz operator look like a star are looked into.