  Let ${\mathcal U}^+$ be the class of analytic functions $f$ such that
$\frac{z}{f(z)}$ has real and positive coefficients and $f^{-1}$ be its
inverse. In this paper we give sharp estimates of the initial coefficients and
initial logarithmic coefficients for $f$, as well as, sharp estimates of the
second and the third Hankel determinant for $f$ and $f^{-1}$. We also show that
the Zalcman conjecture holds for functions $f$ from ${\mathcal U}^+$.
