This paper investigates the precise upper bound of the third-order Hankel determinant for the inverse functions within the function classes $\mathcal S^{*}_{e}$, $\mathcal C_{e}$, $\mathcal S^{*}_{\mathcal L}$, and $\mathcal C_{\mathcal L}$. Additionally, it establishes the upper bound for the second-order Hankel determinant of the logarithmic coefficient in function class $\mathcal S^{*}_{\mathcal L}$, as well as the third-order Hankel determinant for function class $\mathcal C_{car}$.

For some subclasses of Starlike with respect to symmetric points, the sharp upper bounds of the third-order Hankel determinant of inverse functions and Hankel determinants of $ \mathcal H_{2,1}(G_{\Gamma^{-1}}/2) $ and $ \mathcal H_{2,2}(G_{\Gamma^{-1}}/2) $ with logarithmic inverse coefficients are obtained.

For some subclasses of Starlike and Convex functions associated with Gregory coefficient, the sharp upper bounds for the second Hankel determinant and the third-order Hankel determinant and Hankel determinant $ \mathcal H_{2,1}(G_g/2) $ and $ \mathcal H_{2,2}(G_g/2) $ with a logarithmic coeffcient are computed.

In the article, a subclass of Starlike functions linked with Bernoulli,s lemniscate is studied in the unit disc. The bounds of H_3,2 Hankel determinant, the sixth coefficient |b₆| and the Zalcman functionals for the class SL* are obtained. The results are sharp.

In the paper, we investigate the upper bounds of the second and third Hankel determinants for bounded turning functions linked with a strip domain, whose results are sharp. We also obtain the sharp upper bounds of the second-order Hankel determinants of logarithmic coefficients. Finally, we provide the sharp upper bounds of the third-order Hankel determinant of inverse functions, as well as the sharp upper bounds of the second-order Hankel determinants of inverse logarithmic coefficients. All the other results are accurate, except for the estimation of the fourth logarithmic coefficient.

In this paper, we study coefficient problems in some subclasses of convex functions. More precisely, we determine the upper bounds of initial coefficients \( |a_i|(2\leq i\leq 6) \), Zalcman inequalities, the second and third Hankel determinants, the second-order Hankel determinant of logarithmic coefficients and the third Hankel determinant of inverse functions for the class \( \mathcal CL \). All of the bounds are sharp.