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.

Let $S_s^{*}\left(e^z\right)$ denote the class of starlike functions with respect to symmetric points subordinate to the exponential function, i.e., the functions which satisfy in the unit disk $\mathbb{U}$ the condition $\frac{2z{f}^{\prime }\left(z\right)}{f\left(z\right)-f(-z)}\prec e^z(z\in \mathbb{U})$. We obtained the sharp estimate of the second-order Hankel determinants $H_{2,3}\left(f\right)$ and improved the estimate of the third-order $H_{3,1}\left(f\right)$ for this functions class $S_s^{*}\left(e^z\right)$.

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 .

Let R denote the family of analytic and univalent functions f ( z ) in the unit disk Ω = { z ∈ C : | z | < 1} such that Re f ′( z ) > 0. We investigated the Toeplitz determinant for the inverse of the function to f ∈ R and gave the upper bound estimates of the determinants T ₂ (2), T ₂ (3), T ₃ (1) and T ₃ (2) .

In this paper, a new subclass has been defined as $\Omega$ of the univalent function in $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$. The central goal of this paper is to determine estimates for logarithmic coefficients, inverse logarithmic coefficients, some cases of the Hankel determinant and Zalcman functionals $J_{n,m}$ of inverse functions.

Let $\mathcal{B}{T}_{4l}$ denote a subclass of bounded turning functions connected with a four-leaf-type domain. The goal of the study is to probe into the bounds of coefficients $|b_6|,|b_7|,|b_8|$, the bounds of the logarithmic coefficients, and the third-order determinants $|H_{3,1}|,|H_{3,2}|,|H_{3,3}|$ for the functions in this class.