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 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 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. By specializing the sequence, we can obtain an abundant number of sharp inequalities on logarithmic coefficients, some of which are provided in Appendix. 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.

IM Milin在他的1971年的论文中提出了一个不等式的系统，该系统用于复平面单位圆盘上的标准化单价函数的对数系数。这被称为米林猜想，它暗示罗伯逊猜想，而罗伯逊猜想又暗示比伯巴赫猜想。1984年，路易斯·德·布兰吉斯（Louis de Branges）通过展示米林猜想解决了长期存在的比伯巴赫猜想。最近，基于de Branges的证明，O。Roth证明了对数系数的一个有趣的尖锐不等式。在本文中，按照罗斯的思想，我们将展示更多具有凸序列作为权重函数的尖锐不等式。通过对序列进行专门化处理，我们可以得到大量对数系数的尖锐不等式，其中一些在附录中提供。我们还借助线性ODE的de Branges系统的不等式考虑了非凸序列的不等式，其中部分证明由计算机辅助。此外，我们应用了一些不等式来改善先前已知的结果。