A sharp inequality for the logarithmic coefficients of univalent functions

We prove the sharp inequality

$\sum \limits_{k=1}^{\infty} \left( \frac{k}{k+1} \right)^2 \vert ... ...{\infty} \left( \frac{k}{k+1} \right)^2 \frac{1}{k^2}=\frac{2 \, \pi^2-12}{3}$

for the logarithmic coefficients $c_k(f)$ of a normalized univalent function $f$ in the unit disk.