Some important applications of improved Bochner inequality on Finsler manifolds

Abstract: We establish some important inequalities under the condition that the weighted Ricci curvature $\mathrm{Ric}_{\infty}\geq K$ for some constant $K >0$ by using improved Bochner inequality and its integrated form. Firstly, we obtain a sharp Poincar\'{e}-Lichnerowicz inequality. Further, we give a new proof for logarithmic Sobolev inequality. Finally, we obtain an estimate of the volume of geodesic balls.