  In this paper, we define both the upper and lower order of a sense-preserving
harmonic mapping in $\mathbb{D}$. We generalize to the harmonic case some known
results about holomorphic functions with positive lower order and we show some
consequences of a function having finite upper order. In addition, we improve a
related result in the case when there is equality in a known distortion theorem
for harmonic mappings with finite upper order. Some examples are provided to
illustrate the developed theory.
