Radius of starlikeness for some classes containing non-univalent functions

TL;DR

This paper investigates the radius within which certain classes of functions, defined by subordination to specific mappings, are starlike, extending the understanding of non-univalent functions in complex analysis.

Contribution

It determines the sharp radii for classes of functions subordinate to specific mappings, broadening the scope of starlikeness criteria beyond univalent functions.

Findings

Sharp radius bounds for classes subordinate to w_i functions

Extension of starlikeness concepts to non-univalent functions

Characterization of subclasses via subordination conditions

Abstract

A starlike univalent function $f$ is characterized by the function $zf'(z)/f(z)$; several subclasses of these functions were studied in the past by restricting the function $zf'(z)/f(z)$ to take values in a region $\Omega$ on the right-half plane, or, equivalently, by requiring the function $zf'(z)/f(z)$ to be subordinate to the corresponding mapping of the unit disk $\mathbb{D}$ to the region $\Omega$. The mappings $w_1(z):=z+\sqrt{1+z^2}, w_2(z):=\sqrt{1+z}$ and $w_3(z):=e^z$ maps the unit disk $\mathbb{D}$ to various regions in the right half plane. For normalized analytic functions $f$ satisfying the conditions that $f(z)/g(z), g(z)/zp(z)$ and $p(z)$ are subordinate to the functions $w_i, i=1,2,3$ in various ways for some analytic functions $g(z)$ and $p(z)$, we determine the sharp radius for them to belong to various subclasses of starlike functions.