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

Abstract

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