  We prove the concavity of $p$-R\'enyi entropy power for positive solutions to
the doubly nonlinear diffusion equations on $\mathbb{R}^n$ 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 $\mathbb{R}^n$. Moreover, two improvement of
$L^p$-Gagliardo-Nirenberg inequalities are derived.
