Aims: To introduce and investigate a new generalized family of integral operators for analytic functions in the open unit disk, which unifies and extends several known operators by incorporating multiple functions and parameters. Study Design: Mathematical analysis and theoretical development. Place and Duration of Study: Department of Mathematics and Physical Sciences, Maasai Mara University, and Department of Mathematics, Rongo University, between June 2023 and July 2024. Methodology: We define a new integral operator that generalizes the operator studied by Eljamal et al. by incorporating multiple analytic functions and additional complex parameters. This framework provides greater flexibility and subsumes many previously studied integral operators as special cases. Using the theory of subordination, properties of starlike functions, and key lemmas on differential inequalities, we systematically investigate the mapping properties of functions belonging to specific subclasses under this new operator. Results: We establish precise conditions under which the new integral operator preserves the class of starlike functions and the class of starlike functions of order α. Furthermore, we determine its relationship with the Janowski class P(A,B) and provide sharp coefficient estimates. Several illustrative examples are provided to demonstrate the main findings, andconnections with existing results in the literature are highlighted. Conclusion: The new generalized integral operator successfully unifies and extends many known integral operators from the literature. The mapping properties established for starlike, starlike of order α, and Janowski classes provide a solid foundation for further research into convexity, close-to-convexity, and applications to differential subordination.
Wanjala et al. (Thu,) studied this question.