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.

$\mathcal{L}etN$ be the class of functions that convex in one direction and $\mathcal{M}$ denote the class of functions $z{f}^{\prime }\left(z\right)$, where $f\in \mathcal{N}$. In the paper, the third-order Hankel determinants for these classes are estimated. The estimates of ${\mathcal{H}}_{3,1}\left(f\right)$ obtained in the paper are improved.

In this paper, we introduce a new class of generalized close-to-convex functions, which are defined by quasi-subordination relationship. The integral expression, the estimation of the first two terms and the Fekete-Szegö problem of functions belonging to the class are obtained. Our results extend and unify previous work on starlike, convex, and close-to-convex functions under quasi-subordination. Examples are provided to demonstrate sharpness.

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 this paper, we introduce a new generalized differential operator using a new generalized quasi-Hadamard product, and certain new classes of analytic functions using subordination. We obtain certain results concerning the closure properties of the generalized quasi-Hadamard products and the generalized differential operators for this new subclasses of analytic functions with negative and missing coefficients.