# Coefficient and Fekete-Szeg\"o problem estimates for certain subclass of analytic and bi-univalent functions

## TL;DR

This paper investigates coefficient bounds and Fekete-Szeg"o problem estimates for a specific subclass of analytic and bi-univalent functions defined by a particular real part condition involving the function's derivative, expanding understanding of these function classes.

## Contribution

It introduces a new subclass of analytic and bi-univalent functions based on a real part inequality and derives coefficient bounds and Fekete-Szeg"o estimates for this class.

## Key findings

- Derived upper bounds for initial coefficients of the subclass.
- Established Fekete-Szeg"o problem estimates for the subclass.
- Introduced a new subclass of bi-univalent functions with specific geometric properties.

## Abstract

In this paper, we obtain the Fekete-Szeg\"{o} problem for the $k$-th $(k\geq1)$ root transform of the analytic and normalized functions $f$ satisfying the condition \begin{equation*} 1+\frac{\alpha-\pi}{2 \sin \alpha}< {\rm Re}\left\{\frac{zf'(z)}{f(z)}\right\} < 1+\frac{\alpha}{2\sin \alpha} \quad (|z|<1), \end{equation*} where $\pi/2\leq \alpha<\pi$. Afterwards, by the above two-sided inequality we introduce and investigate a certain subclass of analytic and bi-univalent functions in the disk $|z|<1$ and obtain upper bounds for the first few coefficients and Fekete-Szeg\"{o} problem for functions belonging to this analytic and bi-univalent function class.