How do I show that the map is orientation preserving?

Question

Let $f:\mathbb{S}^1\to \mathbb{S}^1$ be an orientation preserving homeomorphism, i.e., the lift $\tilde{f}:\mathbb{R}\to \mathbb{R}$ is monotonic increasing. Now I define $F:\overline{\mathbb{D}}\to \overline{\mathbb{D}}$ by $$F(z)=\begin{cases}\dfrac{1}{2\pi}\int_0^{2\pi}\dfrac{1-|z|^2}{|e^{it}-z|^2}f(e^{it})~dt, & |z|<1 \\f(z), & |z|=1.\end{cases}$$ Then it is not difficult to check that $F$ is harmonic in $\mathbb{D}$ and continuous in $\overline{\mathbb{D}}$. Now how do I show that $\color{red}{F \textrm{ is also an orientation preserving map}?}$