Estimates for Coefficients of Certain Analytic Functions

Abstract

For $-1\le B\le 1$ and $A>B$, let ${\mathcal{S}}^{\ast }[A,B]$ denote the class of generalized Janowski starlike functions consisting of all normalized analytic functions $f$ defined by the subordination $z{f}^{\prime }(z)/f(z)\prec (1+Az)/(1+Bz)$ $(|z|<1)$. For $-1\le B\le 1<A$, we investigate the inverse coefficient problem for functions in the class ${\mathcal{S}}^{\ast }[A,B]$ and its meromorphic counter part. Also, for $-1\le B\le 1<A$, the sharp bounds for first five coefficients for inverse functions of generalized Janowski convex functions are determined. A simple and precise proof for inverse coefficient estimations for generalized Janowski convex functions is provided for the case $A=2\beta -1$ $(\beta >1)$ and $B=1$. As an application, for $F:=f^{-1}$, $A=2\beta -1$ $(\beta >1)$ and $B=1$, the sharp coefficient bounds of $F/F^{\prime }$ are obtained when $f$ is a generalized Janowski starlike or generalized Janowski convex function. Further, we provide the sharp coefficient estimates for inverse functions of normalized analytic functions $f$ satisfying $f^{\prime }(z)\prec (1+z)/(1+Bz)$ $(|z|<1,-1\le B<1)$.