Cosmic constraint on massive neutrinos in viable f(R) gravity with
  producing LCDM background expansion
  Tensions between several cosmic observations were found recently, such as the
inconsistent values of $H_{0}$ (or $\sigma_{8}$) were indicated by the
different cosmic observations. Introducing the massive neutrinos in
$\Lambda$CDM could potentially solve the tensions. Viable $f(R)$ gravity
producing $\Lambda$CDM background expansion with massive neutrinos is
investigated in this paper. We fit the current observational data: Planck-2015
CMB, RSD, BAO and SNIa to constrain the mass of neutrinos in viable $f(R)$
theory. The constraint results at 95\% confidence level are: $\Sigma
m_\nu<0.202$ eV for the active neutrino case, $m_{\nu, sterile}^{eff}<0.757$ eV
with $N_{eff}<3.22$ for the sterile neutrino case. For the effects by the mass
of neutrinos, the constraint results on model parameter at 95\% confidence
level become $f_{R0}\times 10^{-6}> -1.89$ and $f_{R0}\times 10^{-6}> -2.02$
for two cases, respectively. It is also shown that the fitting values of
several parameters much depend on the neutrino properties, such as the cold
dark matter density, the cosmological quantities at matter-radiation equality,
the neutrino density and the fraction of baryonic mass in helium. At last, the
constraint result shows that the tension between direct and CMB measurements of
$H_0$ gets slightly weaker in the viable $f(R)$ model than that in the base
$\Lambda$CDM model.
