# Sharp inequalities for logarithmic coefficients and their applications

## TL;DR

This paper extends sharp inequalities for logarithmic coefficients of univalent functions, generalizing recent results and applying them to improve classical bounds, with proofs involving convex and non-convex sequences and computational assistance.

## Contribution

It introduces more general sharp inequalities for logarithmic coefficients using convex and non-convex sequences, advancing the understanding of their bounds and applications.

## Key findings

- Established new sharp inequalities for logarithmic coefficients.
- Improved bounds on classical univalent function conjectures.
- Applied inequalities to derive stronger results in geometric function theory.

## Abstract

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.