On Coefficient Estimates of Negative Powers and Inverse Coefficients for Certain 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 $\Sigma^*(A,B)$.