  We present the results for a perturbative determination of mass dependent
improvement coefficients $\nu$, $r_s$, $c_E$ and $c_B$ in a relativistic heavy
quark action, which we have designed to control $m_Qa$ errors by extending the
on-shell $O(a)$ improvement program to the case of $m_Q \gg \Lambda_{\rm QCD}$,
where $m_Q$ is the heavy quark mass. The parameters $\nu$ and $r_s$ are
determined from the quark propagator and $c_E$ and $c_B$ are from the on-shell
quark-quark scattering amplitude. We show that all the parameters, together
with the quark wave function and the mass renormalization factors, are
determined free from infrared divergences once their tree level values are
properly tuned. Results of these parameters are shown as a function of $m_Q a$
for various improved gauge actions.
