Coefficients of the Inverse Functions and Radius Estimates of Certain Starlike Functions

TL;DR

This paper derives optimal bounds on coefficients, Fekete-Szeg"o functional, and Hankel determinants for inverse functions of Ma-Minda starlike functions, along with radius estimates for specific subclasses.

Contribution

It provides the first comprehensive bounds on inverse function coefficients and radius estimates within the Ma-Minda class of starlike functions.

Findings

Optimal bounds on second and third inverse coefficients.

Bounds on Fekete-Szeg"o functional and second Hankel determinant.

Radius estimates for particular subclasses.

Abstract

Ma-Minda class (of starlike functions) consists of all normalized analytic functions $f$ on the unit disk for which the image of $zf'(z)/f(z)$ is contained in the some starlike region in the right-half plane. We obtain the best possible bounds on the second and third coefficient for the inverse functions of functions in the Ma-Minda class. The bounds on the Fekete-Szeg\"o functional and the second Hankel determinant of the inverse functions of the functions belonging to the Ma-Minda class are also determined. Further, the bounds on the first five coefficients of the inverse functions are investigated for two particular subclasses of the Ma-Minda class. In addition, some radius estimates associated with the two subclasses are also computed.