  In our prior papers, we considered the non-relativistic linear stability
analysis of magnetized jets that do not have current sheet at the boundary. In
this paper, we extend our analysis to relativistic jets. In order to find the
unstable modes of current sheet-free, magnetized relativistic jets, we
linearize full relativistic magnetohydrodynamics equations and solve them
numerically. We find the dispersion relation of the pinch and kink mode
instabilities. By comparing the dispersion relations of mildly relativistic jet
(Lorentz factor 2) with moderately relativistic jet (Lorentz factor 10), we
find that the jet with higher Lorentz factor is significantly more stable in
both pinch and kink modes. We show that inclusion of the current sheet-free
magnetic field in the jet further enhances the stability. Both pinch and kink
mode instabilities become progressively more stable with increasing
magnetization. We also show a scaling relation between the maximum temporal
growth rate of the unstable mode and the Lorentz factor of the jet. The maximum
temporal growth rates of the unstable modes are inversely proportion to the
Lorentz factors for most of the modes that we study. However, for the
fundamental pinch mode it is inversely proportional to the square of the
Lorentz factor. This very beneficial scaling relation holds regardless of the
presence of a magnetic field.
