Coefficient Estimates of a New Subclass of <i>r</i>-Fold Symmetric Bi-Univalent Functions Involving Ruscheweyh Derivative

In this work, the authors introduce and study a new subclass ofMh,uΣr(γ,η,κ,λ), which is r-fold bi-univalent functions in open-unit disk 𝔄. Using the Rushwaya derivative operator, the study investigates the properties and behavior of an analytically generalized r-fold symmetric two-monovalent function in the open-unit disk 𝔄. The analysis calculates the Taylor-Maclarin coefficients |ar+1| and |a2r+1|, which represent one of its key terms these estimates Mh,uΣr(γ,η,κ,λ). Subclass of functions important for understanding analytical and geometric behavior they provide valuable insights into the development of these concepts and their relationship to the geometric properties of the functions. Furthermore, research goes beyond coefficient estimates to examine the significance of results. By analyzing this expression, the study highlights the circumstantial importance of the Mh,uΣr(γ,η,κ,λ) subclass plant. The importance of this new subclass is essentially its ability to add special information to specific operator users, as well as the inclusion of multiple values to increase its flexibility and robustness. Furthermore, by linking us to prior research in the field, the study strengthens the theoretical framework and deepens our understanding of these activities, thus contributing to extensive knowledge in this area.