  We investigate predictions on the triple Higgs boson couplings with radiative
corrections in the model with an additional real singlet scalar field. In this
model, the second physical scalar state ($H$) appears in addition to the Higgs
boson ($h$) with the mass 125 GeV. The $hhh$ vertex is calculated at the
one-loop level, and its possible deviation from the predictions in the standard
model is evaluated under various theoretical constraints. The decay rate of $H
\to hh$ is also computed at the one-loop level. We also take into account the
bound from the precise measurement of the $W$ boson mass, which gives the upper
limit on the mixing angle $\alpha$ between two physical Higgs bosons for a
given value of the mass of $H$ ($m_H^{}$). We find that the deviation in the
$hhh$ coupling from the prediction in the standard model can maximally be about
250\%, 150\% and 75\% for $m_H^{}=300$, 500 and 1000 GeV, respectively, under
the requirement that the cutoff scale of the model is higher than 3 TeV. We
also discuss deviations from the standard model prediction in double Higgs
boson production from the gluon fusion at the LHC using the one-loop corrected
Higgs boson vertices.
