Coefficients of the inverse of functions for the subclass of $\mathcal U (\lambda)$

TL;DR

This paper establishes sharp bounds for the first three coefficients of inverse functions within a specific subclass of analytic functions, extending understanding of their coefficient behavior under certain subordination conditions.

Contribution

It provides the first sharp coefficient bounds for the inverse functions of a subclass of $ ext{U}( ext{lambda})$ functions defined by a particular subordination.

Findings

Sharp bounds for the first three inverse coefficients are derived.

The bounds are optimal and attained under specific extremal functions.

The results extend existing coefficient estimates for subclasses of univalent functions.

Abstract

Let ${\mathcal A}$ be the class of functions $f$ that are analytic in the unit disk ${\mathbb D}$ and normalized such that $f(z)=z+a_2z^2+a_3z^3+\cdots$. Let $0<\lambda\le1$ and \[ {\mathcal U}(\lambda) = \left\{ f\in{\mathcal A}: \left |\left (\frac{z}{f(z)} \right )^{2}f'(z)-1\right | < \lambda,\, z\in{\mathbb D} \right\}. \] In this paper sharp upper bounds of the first three coefficients of the inverse function $f^{-1}$ are given in the case when \[\frac{f(z)}{z}\prec \frac{1}{(1-z)(1-\lambda z)}.\]