Sharp inequalities for logarithmic coefficients and their applications

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