A sharp inequality for the logarithmic coefficients of univalent functions

Abstract:

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