On Coefficient Estimates of Negative Powers and Inverse Coefficients for Certain Starlike Functions

TL;DR

This paper investigates coefficient estimates for certain starlike functions and their inverses, focusing on negative powers and inverse coefficients within specific classes of analytic and meromorphic functions.

Contribution

It provides new bounds for the coefficients of functions in the classes al{S}^*(A,B) and al{ extSigma}^*(A,B), including their inverses, for negative powers.

Findings

Derived bounds for coefficients of (f(z)/z)^{-mbda} for functions in al{S}^*(A,B)

Established coefficient estimates for inverses of functions in al{S}^*(A,B) and al{ extSigma}^*(A,B)

Extended classical results to broader classes of starlike functions

Abstract

For $-1\le B<A\le 1$, let $\mathcal{S}^*(A,B)$ denote the class of normalized analytic functions $f(z)= z+\sum_{n=2}^{\infty}a_n z^n$ in $|z|<1$ which satisfy the subordination relation $zf'(z)/f(z)\prec (1+Az)/(1+Bz)$ and $\Sigma^*(A,B)$ be the corresponding class of meromorphic functions in $|z|>1$. For $f\in\mathcal{S}^*(A,B)$ and $\lambda>0$, we shall estimate the absolute value of the Taylor coefficients $a_n(-\lambda,f)$ of the analytic function $(f(z)/z)^{-\lambda}$. Using this we shall determine the coefficient estimate for inverses of functions in the classes $\mathcal{S}^*(A,B)$ and $\Sigma^*(A,B)$.