# Starlikeness Associated With The Exponential Function

## TL;DR

This paper explores the class of functions related to the exponential function within complex analysis, providing new conditions for functions to be considered starlike associated with the exponential, expanding understanding of differential subordinations.

## Contribution

It investigates properties of admissible functions linked to the exponential function and derives new sufficient conditions for functions to be starlike related to the exponential.

## Key findings

- Derived new sufficient conditions for exponential starlikeness
- Extended differential subordination theory to exponential functions
- Provided applications to normalized analytic functions

## Abstract

Given a domain $\Omega$ in the complex plane $\mathbb{C}$ and a univalent function $q$ defined in an open unit disk $\mathbb{D}$ with nice boundary behaviour, Miller and Mocanu studied the class of admissible functions $\Psi(\Omega,q)$ so that the differential subordination $\psi(p(z),zp(z),z^2p''(z);z)\prec h(z)$ implies $p(z)\prec q(z)$ where $p$ is an analytic function in $\mathbb{D}$ with $p(0)=1$, $\psi:\mathbb{C}^3\times \mathbb{D}\to\mathbb{C}$ and $\Omega=h(\mathbb{D})$. This paper investigates the properties of this class for $q(z)=e^z$. As application, several sufficient conditions for normalized analytic functions $f$ to be in the subclass of starlike functions associated with the exponential function are obtained.