My Try:
I did part a,b and c. Having trouble with d. Followed the hint and got $\displaystyle f(z)-RHS=\frac{1}{2\pi i}\Bigg[\int_\Gamma \frac{f(\zeta)}{(\zeta - z)}\Bigg(1-\frac{w(z)}{w(\zeta)}\Bigg)d\zeta\Bigg]$. Then I wanted to diffetentiate this $(n+1)$ times with respect to $z$ and get $0$, but failed. Can anybody please help me to proceed?