I think I am supposed to use Rouche Theorem and Cauchy Argument Principle for this problem as well:
Any ideas? Any help is appreciated! thanks
Define $\displaystyle g(z) = f(z) - z$ and use Rouche's theorem by showing that $\displaystyle g(z)$ has a zero in the disk. That will mean there is $\displaystyle a\in D(0,1)$ such that $\displaystyle g(a)=0\implies f(a) = a$ and hence $\displaystyle f$ has a fixed point.