We prove the concavity of p-Rényi entropy power for positive solutions to the doubly nonlinear diffusion equations on R n or compact Riemannian manifolds with nonnegative Ricci curvature. As applications, we give new proofs of the sharp L p -Sobolev inequality and L p -Gagliardo-Nirenberg inequalities on R n . Moreover, two improvement of L p -Gagliardo-Nirenberg inequalities are derived. (2010). Primary 58J35, 35K92; Secondary 35B40,35K55
Mathematics Subject Classification