# Estimates for Coefficients of Certain Analytic Functions

## TL;DR

This paper derives sharp coefficient bounds for inverse functions within classes of generalized Janowski starlike and convex functions, expanding understanding of their analytic and meromorphic properties.

## Contribution

It provides new inverse coefficient estimates for generalized Janowski functions and their meromorphic counterparts, including sharp bounds for initial coefficients and specific function ratios.

## Key findings

- Sharp bounds for first five inverse coefficients of Janowski convex functions.
- Coefficient estimates for ratios of inverse functions like F/F' in these classes.
- Simplified proofs for inverse coefficient bounds in special cases.

## Abstract

For $ -1 \leq B \leq 1$ and $A>B$, let $\mathcal{S}^*[A,B]$ denote the class of generalized Janowski starlike functions consisting of all normalized analytic functions $f$ defined by the subordination $z f'(z)/f(z) \prec (1+ A z)/(1+ B z)$ $(|z|<1)$. For $-1 \leq B \leq 1<A$, we investigate the inverse coefficient problem for functions in the class $\mathcal{S}^*[A,B]$ and its meromorphic counter part. Also, for $ -1 \leq B \leq 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 $F/F'$ 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 $f'(z) \prec (1+z)/(1+B z)$ $(|z|<1, -1 \leq B < 1)$.