  We prove that convex functions of finite order on the real line and
subharmonic functions of finite order on finite dimensional real space, bounded
from above outside of some set of zero relative Lebesgue density, are bounded
from above everywhere. It follows that subharmonic functions of finite order on
the complex plane, entire and plurisubharmonic functions of finite order, and
convex or harmonic functions of finite order bounded from above outside some
set of zero relative Lebesgue density are constant.
