Radii of Starlikeness and Convexity of Analytic Functions Satisfying Certain Coefficient Inequalities

TL;DR

This paper determines the exact radii of starlikeness and convexity for certain analytic functions with coefficient constraints, extending understanding of geometric function theory.

Contribution

It provides sharp radii results for functions with specific coefficient inequalities, including a class related to Carathéodory functions, advancing geometric function theory.

Findings

Sharp radii of starlikeness and convexity obtained

Results for functions with coefficients bounded by n or M/n

Includes analysis of functions related to Carathéodory functions

Abstract

For $0\leq \alpha <1$, the sharp radii of starlikeness and convexity of order $\alpha$ for functions of the form $f(z)=z+a_2z^2+a_3z^3+...$ whose Taylor coefficients $a_n$ satisfy the conditions $|a_2|=2b$, $0\leq b\leq 1$, and $|a_n|\leq n $, $M$ or $M/n$ ($M>0$) for $n\geq 3$ are obtained. Also a class of functions related to Carath\'eodory functions is considered.