Lucas-Balancing Polynomials Within <em>ρ</em>-bi-Pseudo-Starlike Functions: Analtycical Estimate

In this study, we introduce a novel class of \rho-bi-pseudo-starlike functions, constructed based on symmetric points associated with the boundary constraints of the initial Taylor-Maclaurin coefficients. A comprehensive analysis of the Fekete-Szegö problem is conducted for functions belonging to this class. Furthermore, we identify special cases of our main ndings, demonstrating signicant improvements in the estimation of the initial Taylor-Maclaurin coefficients. The results presented in this work contribute to the broader understanding of bi-univalent function theory and provide new insights into function coefficient bounds in complex analysis.