We study the convergence of states under continuous-time depolarizing channels with full rank fixed points in terms of the relative entropy. The optimal exponent of an upper bound on the relative entropy in this case is given by the log-Sobolev-1 constant. Our main result is the computation of this constant. As an application, we use the log-Sobolev-1 constant of the depolarizing channels to improve the concavity inequality of the von Neumann entropy. This result is compared to similar bounds obtained recently by Kim and we show a version of Pinsker’s inequality, which is optimal and tight if we fix the second argument of the relative entropy. Finally, we consider the log-Sobolev-1 constant of tensor-powers of the completely depolarizing channel and use a quantum version of Shearer’s inequality to prove a uniform lower bound