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In the present paper, we first investigate a new locally constrained mean curvature flow (1.7) for starshaped hypersurfaces in hyperbolic space H n+1 and prove its longtime existence, exponential convergence. As an application, we establish a new sharp Michael-Simon inequality for mean curvatures inprovided that M is starshaped and f is a positive smooth function, where λ ′ (r) = cosh r. In particular, when f is of constant, (0.1) is exactly the Minkowski type inequality established by Brendle, Hung and Wang in [7] for mean convex and starshaped hypersurfaces.In the second part of this paper, we use a locally constrained inverse curvature flow (1.10) in H n+1 , which was introduced by Scheuer and Xia [30] to establish a new sharp Michael-Simon inequality for k-th mean curvatures of starshaped and strictly k-convex domain

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