Third-order Hankel determinant sharp estimates for the inverse of complex valued holomorphic functions

Determining coefficient estimates of inverse complex valued function is often more challenging than determining the function itself, leading to a lack of articles on inverse functions by examining sharp estimates of the third-order Hankel determinant. This article uses Carathéodory functions to estimate the initial coefficients related to problems for inverse of functions that are bounded turning and subordinated with the cosine function. It covers some inverse coefficients problems together with the second and third-order Hankel determinants. All the estimations that were found are sharp and all the extremal functions and their inverses are also provided.