= Solution
Choose a small positively oriented circle $|w|=\rho$ in $U$ within the convergence disk of the inverse series. Cauchy's coefficient formula gives
$$
c_n=\frac1{2\pi i}\int_{|w|=\rho}\frac{g(w)}{w^{n+1}}dw.
$$
Let $\gamma=g(\{|w|=\rho\})$. The <biholomorphism> property makes it a simple positively oriented closed curve about zero. Under $w=f(z)$, $g(w)=z$ and $dw=f'(z)dz$, so
$$
c_n=\frac1{2\pi i}\int_\gamma\frac{zf'(z)}{f(z)^{n+1}}dz=\frac1{2\pi in}\int_\gamma\frac{dz}{f(z)^n}.
$$
The last equality is (a). Apply (b), giving
$$
\boxed{c_n=\frac1{n!}\left[\frac{d^{n-1}}{dz^{n-1}}\left(\frac z{f(z)}\right)^n\right]_{z=0}.}
$$
This is the local <Lagrange inversion theorem>; the <simple zero> ensures that the expression being differentiated has a removable singularity and a nonzero value at zero.
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