Starlikeness of Analytic Functions with Subordinate Ratios

TL;DR

This paper determines the radius of starlikeness for a class of normalized analytic functions subordinate to specific functions, expanding understanding of geometric properties in complex analysis.

Contribution

It introduces new radius results for functions subordinate to $\,h(z)=\, ext{either}\,\sqrt{1+z}\, ext{or}\,e^z$, including $\, ext{G}$-radius for various starlike subclasses.

Findings

Radius of starlikeness for the class with $h(z)=\, ext{either}\,\sqrt{1+z}\, ext{or}\,e^z$

G-radius for Janowski and parabolic starlike functions

Explicit bounds for the subclasses' geometric properties

Abstract

Let $h$ be a non-vanishing analytic function in the open unit disc with $h(0)=1$. Consider the class consisting of normalized analytic functions $f$ whose ratios $f(z)/g(z)$, $g(z)/z p(z)$, and $p(z)$ are each subordinate to $h$ for some analytic functions $g$ and $p$. The radius of starlikeness is obtained for this class when $h$ is chosen to be either $h(z)=\sqrt{1+z}$ or $h(z)=e^z$. Further $\mathcal{G}$-radius is also obtained for each of these two classes when $\mathcal{G}$ is a particular widely studied subclass of starlike functions. These include $\mathcal{G}$ consisting of the Janowski starlike functions, and functions which are parabolic starlike.