The Brownian motion of a particle in a one-dimensional periodic potential subjected to a uniform external force F is studied. Using the formula for the diffusion coefficient D obtained by other authors and an alternative one derived from the Fokker-Planck equation in the present work, D is compared with the differential mobility μ = dv/dF where v is the average velocity of the particle. Analytical and numerical calculations indicate that inequality D >μ k_BT, with k_B the Boltzmann constant and

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... T the temperature, holds if the periodic potential is symmetric, while it is violated for asymmetric potentials when F is small but nonzero.