Sharp Estimates for the Logarithmic Coefficients, Fekete-Szegö Functional, and Third Hankel Determinant of the Inverse k-th Root Transformation for a Generalized Class of Bounded Boundary Rotation

Résumé

One of the central challenges in Geometric Function Theory is the determination of coefficient estimates for the inverses of holomorphic mappings. The present research proposes a novel generalized subclass of analytic functions, symbolized by , which is characterized by a linear combination involving the first and second derivatives: . We explore the geometric properties of the inverse function corresponding to the -th root transformation for functions within this class. The primary objective is to deduce the sharp upper limit for the third-order Hankel determinant . Additionally, to ensure a comprehensive study, we derive sharp bounds for the logarithmic coefficients, resolve the FeketeSzegö problem, and present a numerical validation to visually corroborate the theoretical findings. The outcomes demonstrate that incorporating the second derivative parameter substantially improves the coefficient bounds in comparison to the classical functions of bounded turning.