Third Hankel determinant for the inverse of reciprocal of bounded turning functions has a positive real part of order alpha

Let $RT$ be the class of functions $f(z)$ univalent in the unit disk $E = {z : |z| < 1}$ such that $\mathrm{Re}\, f'(z) > 0$, $z\in E$, and $H_3(1)$ be the third Hankel determinant for inverse function to $f(z)$. In this paper we obtain, first an upper bound for the second Hankel determinant, $|t_2 t_3 - t_4|$, and the best possible upper bound for the third Hankel determinant $H3(1)$ for the functions in the class of inverse of reciprocal of bounded turning functions having a positive real part of order alpha.