Directional Coefficient Inequalities for Strongly Starlike Mappings in Complex Banach Spaces

We study a class of strongly starlike mappings of order $\alpha$ on the open unit ball of a complex Banach space under a scalar-profile condition. Using a slice reduction along one-dimensional complex subspaces, we prove a coefficient correspondence theorem identifying the directional coefficients with the Taylor coefficients of an associated one-variable strongly starlike function. We further prove a lifting theorem showing that every one-variable strongly starlike function can be realized by a scalar-profile Banach-space mapping. Consequently, the extremal problems for polynomial directional coefficient functionals coincide exactly with their one-variable counterparts. We apply this correspondence to obtain bounds for $\mathrm{\mid }{A}_2{A}_3-A_4\mathrm{\mid }$, $\mathrm{\mid }{A}_2{A}_4-A_3^2\mathrm{\mid }$, and $\mathrm{\mid }{A}_3-A_2^2\mathrm{\mid }$, together with the corresponding estimates for $\mathrm{\mid }{A}_4\mathrm{\mid }$ and $\mathrm{\mid }{A}_5\mathrm{\mid }$. We also establish a parameterized Fekete–Szegő estimate and a sharp bound for the second Hankel determinant of directional logarithmic coefficients.