A sharp inequality for the logarithmic coefficients of univalent functions

We prove the sharp inequality \[ ∑ k = 1 ∞ ( k k + 1 ) 2 | c k ( f ) | 2 ≤ 4 ∑ k = 1 ∞ ( k k + 1 ) 2 1 k 2 = 2 π 2 − 12 3 \sum \limits _{k=1}^{\infty } \left ( \frac {k}{k+1} \right )^2 |c_k(f)|^2 \le 4 \sum _{k=1}^{\infty } \left ( \frac {k}{k+1} \right )^2 \frac {1}{k^2}=\frac {2 \, \pi ^2-12}{3} \] for the logarithmic coefficients c k ( f ) c_k(f) of a normalized univalent function f f in the unit disk.