  Taking into account the mass splittings between three active neutrinos, we
investigate impacts of dark energy on constraining the total neutrino mass
$\sum m_{\nu}$ by using recent cosmological observations. We consider two
typical dark energy models, namely, the $w$CDM model and the holographic dark
energy (HDE) model, which both have an additional free parameter compared with
the $\Lambda$CDM model. We employ the Planck 2015 data of CMB temperature and
polarization anisotropies, combined with low-redshift measurements on BAO
distance scales, type Ia supernovae, Hubble constant, and Planck lensing.
Compared to the $\Lambda$CDM model, our study shows that the upper limit on
$\sum m_{\nu}$ becomes much looser in the $w$CDM model while much tighter in
the HDE model. In the HDE model, we obtain the $95\%$ CL upper limit $\sum
m_{\nu}<0.105~\textrm{eV}$ for three degenerate neutrinos. This might be the
most stringent constraint on $\sum m_{\nu}$ by far and is almost on the verge
of diagnosing the neutrino mass hierachies in the HDE model. However, the
difference of $\chi^2$ is still not significant enough to distinguish the
neutrino mass hierarchies, even though the minimal $\chi^2$ of the normal
hierarchy is slightly smaller than that of the inverted hierarchy.
