Spectral Improvements of Geometric Inequalities on Closed Kähler Manifolds

Let $(M,g,J)$ be a closed Kähler manifold satisfying $\mathrm{Ric}⩾g$. We establish improved Liouville theorems for the Euler–Lagrange equations associated with the Beckner–Sobolev inequalities by incorporating the first positive eigenvalue of the $\bar{\mathrm{\partial }}$-Laplacian into a differential-identity argument. As a consequence, we obtain improved Sobolev and Beckner inequalities that refine the known Riemannian and Kähler estimates when the first eigenvalue is sufficiently large. We also derive new upper bounds for the diameter of $(M,g)$.