# Improved Beckner-Sobolev inequalities on K\"ahler manifolds

## TL;DR

This paper establishes new Beckner-Sobolev inequalities on compact Kähler manifolds with positive Ricci curvature, leading to improved geometric bounds such as a tighter diameter estimate.

## Contribution

It introduces novel Beckner-Sobolev inequalities specific to Kähler manifolds with positive Ricci curvature, enhancing existing geometric analysis tools.

## Key findings

- New Beckner-Sobolev inequalities on Kähler manifolds
- Improved diameter bounds surpassing Bonnet-Myers estimate
- Applications to geometric analysis and curvature estimates

## Abstract

We prove new Beckner-Sobolev type inequalities on compact K\"{a}hler manifolds with positive Ricci curvature. As an application, we obtain a diameter upper bound that improves the Bonnet-Myers bound.