  I. M. Milin proposed, in his 1971 paper, a system of inequalities for the
logarithmic coefficients of normalized univalent functions on the unit disk of
the complex plane. This is known as the Lebedev-Milin conjecture and implies
the Robertson conjecture which in turn implies the Bieberbach conjecture. In
1984, Louis de Branges settled the long-standing Bieberbach conjecture by
showing the Lebedev-Milin conjecture. Recently, O.~Roth proved an interesting
sharp inequality for the logarithmic coefficients based on the proof by de
Branges. In this paper, following Roth's ideas, we will show more general sharp
inequalities with convex sequences as weight functions and then establish
several consequences of them. We also consider the inequality with the help of
de Branges system of linear ODE for non-convex sequences where the proof is
partly assisted by computer. Also, we apply some of those inequalities to
improve previously known results.
