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

Some important applications of improved Bochner inequality on Finsler manifolds