Analytic and geometric properties of open door functions

TL;DR

This paper explores the analytic and geometric characteristics of solutions to a specific differential equation involving an analytic function, aiming to understand conditions that ensure starlikeness of related functions within the unit disk.

Contribution

It introduces new insights into the properties of open door functions and establishes bounds for conditions implying starlikeness of analytic functions.

Findings

Determined the largest constant c for starlikeness conditions.

Analyzed the geometric behavior of solutions to the differential equation.

Provided criteria linking the imaginary part condition to starlikeness.

Abstract

In this paper, we study analytic and geometric properties of the solution $q(z)$ to the differential equation $q(z)+zq'(z)/q(z)=h(z)$ with the initial condition $q(0)=1$ for a given analytic function $h(z)$ on the unit disk $|z|<1$ in the complex plane with $h(0)=1.$ In particular, we investigate the possible largest constant $c>0$ such that the condition $|\Im[zf"(z)/f'(z)]|<c$ on $|z|<1$ implies starlikeness of an analytic function $f(z)$ on $|z|<1$ with $f(0)=f'(0)-1=0.$