  We carry out the one-loop calculation of mass dependent ${\cal O}(a)$
improvement coefficients in the relativistic heavy quark action recently
proposed, employing the ordinary perturbation theory with the fictitious gluon
mass as an infrared regulator. We also determine renormalization factors and
improvement coefficients for the axial-vector current at the one-loop level. It
is shown that the improvement coefficients are infrared finite at the one-loop
level if and only if the improvement coefficients in the action are properly
tuned at the tree level.
