In the paper, a family of bounded turning functions involving a four-leaf-type domain is studied in the unit disk. The goal of the study is to explore the bounds of second and the third Hankel determinant for functions in the class. All of obtained bounds have been sharp.

In this paper, we first establish a sharp version of Landau-type theorem of polyharmonic mappings. Then, we establish two versions of Landau-type theorems of polyharmonic mappings by applying Cauchy-inequality, which improve the corresponding theorems given in Luo et al.(Computational Methods and Function Theory, 23(2):303-325, 2023). Finally, three new Landau-type theorems of log-pharmonic mappings are established, one of which improves upon a result given in Bai et al. (Complex Analysis and Operator Theory, 13(2):321-340, 2019).

In this paper, we first establish a Landau-Bloch type theorem for certain bounded and normalized biharmonic mappings F(z) = |z|2g(z) + h(z), where g(z) and h(z) are harmonic in the unit disk with |g(z)| <= M1, |h(z)| <= M2. In particular, our result is sharp when M1 = M2 = 1. Then, we establish several new versions of Landau-Bloch type theorems for certain normalized biharmonic mappings with the coefficients condition in place of |h(z)| <= M2 or |g(z)| <= M1, and obtain several sharp results.

The paper introduces and examines a subclass of bounded turning functions associated with a single-lobed elliptic domain. It provides precise bounds for the second and third-order Hankel determinants, the Hankel determinant with logarithmic coefficients, as well as the higher-order Schwarzian derivative. Additionally, the study establishes upper bounds for the corresponding coefficients of the inverse functions.

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.