We clarify inclusion relations between sufficient conditions which guarantee that the equilibrium of the damped harmonic oscillator is uniformly asymptotically stable, where is locally integrable. Those conditions work to suppress the rapid growth of the frictional force expressed by the integral amount of the damping coefficient h. The obtained inclusion relations for uniform asymptotic stability are compared with known inclusion relations for asymptotic stability. Two correlation diagrams are attached to facilitate understanding of the inclusion relations. By giving a concrete example, remaining problems are pointed out.