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

Abstract

For $-1\le B<A\le 1$, let ${\mathcal{S}}^{\ast }(A,B)$ 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 $z{f}^{\prime }(z)/f(z)\prec (1+Az)/(1+Bz)$ and ${\Sigma }^{\ast }(A,B)$ be the corresponding class of meromorphic functions in $|z|>1$. For $f\in {\mathcal{S}}^{\ast }(A,B)$ 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 ${\mathcal{S}}^{\ast }(A,B)$ and ${\Sigma }^{\ast }(A,B)$.

14 pages (To appear)