Suppose that S T 1 is a class of close-to-star functions. In this paper, we investigated the estimate of Zalcman functional on the logarithmic coefficients and the third Hankel determinant for the class S T 1 with the determinant entry of logarithmic coefficients. Also, we obtained the sharp bounds of Zalcman functional J 2 , 4 f and J 3 , 3 f for the class S T 1 .

For starlike, convex and bounded turning functions linked with nephroid function, the sharp upper bounds of the third-order Hankel determinant, the third-order Hankel determinant of inverse functions and the second-order Hankel determinant of logarithmic coefficients are computed.

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.

This paper introduces a novel class of convex functions associated with a strip domain and establishes the upper bounds for the coefficients of initial terms, as well as second and third-order Hankel determinants. It provides exact upper bounds for the third-order Hankel determinants of both the inverse of starlike functions and convex functions, along with the upper bounds for the second-order Hankel determinants of the logarithmic coefficients related to these functions.

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.

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 $\mathcal C_{car}$.