Worked solution: De Branges's proof of the Bieberbach conjecture via Milin's conjecture (1984)

Since $\Phi_n(0)=0$ and Steps 5--6 give $\Phi_n'(t)\le0$, we have $\Phi_n(1)\le0$, hence the full Milin sum is non-positive and the Lebedev--Milin inequality yields $|a_n|\le n$. The argument also identifies equality: equality throughout forces a constant Loewner driving function, corresponding to rotations of the Koebe function.