Some $m$-Fold Symmetric Bi-Univalent Function Classes and Their Associated Taylor-Maclaurin Coefficient Bounds

The Ruscheweyh derivative operator is used in this paper to introduce and investigate interesting general subclasses of the function class $\Sigma_{\mathrm{m}}$ of $m$-fold symmetric bi-univalent analytic functions. Estimates of the initial Taylor-Maclaurin coefficients $\left|a_{m+1}\right|$ and $\left|a_{2 m+1}\right|$ are obtained for functions of the subclasses introduced in this study, and the consequences of the results are discussed. The results presented would generalize and improve on some recent works by many earlier authors. In some cases, our estimates are better than the existing coefficient bounds. Furthermore, within the engineering domain, this paper delves into a series of complex issues related to analytic functions, $m$-fold symmetric univalent functions, and the utilization of the Ruscheweyh derivative operator. These problems encompass a broad spectrum of engineering applications, including the optimization of optical system designs, signal processing for antenna arrays, image compression techniques, and filter design for control systems. The paper underscores the crucial role of these mathematical concepts in addressing practical engineering dilemmas and fine-tuning the performance of various engineering systems. It emphasizes the potential for innovative solutions that can significantly enhance the reliability and effectiveness of engineering applications.