  In this paper, we obtain the upper bounds to the third Hankel determinants
for starlike functions of order $\alpha$, convex functions of order $\alpha$
and bounded turning functions of order $\alpha$. Furthermore, several relevant
results on a new subclass of close-to-convex harmonic mappings are obtained.
Connections of the results presented here to those that can be found in the
literature are also discussed.
