Estimates for Coefficients of Certain Analytic Functions

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 $z f’(z)/f(z) \prec (1+ A z)/(1+ B z)$ $(\mathrm{\mid }z\mathrm{\mid }<1)$. 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 $F\mathrm{/}{F}^{\mathrm{\prime }}$ 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 $f^{\mathrm{\prime }}(z)\prec (1+z)\mathrm{/}(1+Bz)$ $(\mathrm{\mid }z\mathrm{\mid }<1,-1\le B<1)$.