Radius Constants for Analytic Functions with Fixed Second Coefficient

TL;DR

This paper determines the exact radii for Janowski starlikeness and related properties of analytic functions in the unit disk with a fixed second coefficient, considering various bounds on higher coefficients.

Contribution

It provides sharp radius constants for Janowski starlikeness and other properties for functions with fixed second coefficient and specific bounds on higher coefficients.

Findings

Sharp radius of Janowski starlikeness obtained.

Radius constants derived for functions with bounded coefficients.

Results applicable to functions with fixed second coefficient and coefficient bounds.

Abstract

Let $f(z)=z+\sum_{n=2}^{\infty}a_nz^n$ be analytic in the unit disk with second coefficient $a_2$ satisfying $|a_2|=2b$, $0\leq b\leq1$. Sharp radius of Janowski starlikeness and other radius constants are obtained when $|a_n|\leq cn+d$ ($c,d\geq0$) or $|a_n|\leq c/n$ ($c>0$) for $n\geq3$.