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We give several Bishop-Gromov relative volume comparisons with integral Riccicurvature which improve the results in \cite{PW1}. Using one of these volumecomparisons, we derive an estimate for the volume entropy in terms of integralRicci curvature which substantially improves an earlier estimate in \cite{Au2}and give an application on the algebraic entropy of its fundamental group. Wealso extend the almost minimal volume rigidity of \cite{BBCG} to integral Riccicurvature.

Improved relative volume comparison for integral Ricci curvature and applications to volume entropy