# New class of symmetric starlike functions subordinate to the generating function of Gregory coefficients

## TL;DR

This paper introduces a new class of symmetric starlike functions and derives sharp coefficient bounds using subordination and generating functions of Gregory coefficients.

## Contribution

The novelty lies in defining a new family of symmetric starlike functions and providing sharp coefficient bounds using the generating function of Gregory coefficients.

## Key findings

- Sharp bounds for the first five coefficients of symmetric starlike functions are established.
- The Fekete-Szego problem and the Hankel determinant of order three are solved for the defined function class.
- Optimal bounds for logarithmic and inverse functions within the class are derived.

## Abstract

Function theory research has long struggled with the challenge of deriving sharp estimates for the coefficients of analytic and univalent functions. Researchers have advanced the field by developing and applying a variety of approaches to get these bounds. In the current paper, we apply the technique of subordination, we define the family of symmetric starlike functions which is related to generating function of Gregory coefficients. We provide sharp bounds for the problem concerning the coefficients of the family of symmetric starlike functions connected to the generating function of Gregory coefficients by utilizing the notion of functions with positive real component. These problems include first five sharp coefficient bounds and Fekete-Szego problem along with the Hankel determinant of order three. Additionally, we explore the optimal bounds (sharp bounds) for two important functions, the logarithmic function and the inverse function within the same class of symmetric starlike functions which is related to generating function of Gregory coefficients.