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

Abstract

Let $\mathcal{A}$ be the class of functions $f$ that are analytic in the unit disk $\mathbb{D}$ and normalized such that $f(z)=z+a_2{z}^2+a_3{z}^3+\cdots$. Let $0<\lambda \le 1$ and \[ {\mathcal U}(λ) = \left\{ f\in{\mathcal A}: \left |\left (\frac{z}{f(z)} \right )^{2}f'(z)-1\right | < λ,\, z\in{\mathbb D} \right\}. \] In this paper sharp upper bounds of the first three coefficients of the inverse function $f^{-1}$ are given in the case when \[\frac{f(z)}{z}\prec \frac{1}{(1-z)(1-λz)}.\]