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

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 $$\frac{f(z)}{z}\prec \frac{1}{(1-z)(1-\lambda z)}.$$