  We present an analysis of the heavy quark structure functions from the
$k_{t}$ factorization scheme, using unifying the color dipole picture and
double asymptotic scaling approaches at small $x$. The gluon distribution is
obtained from the Golec-Biernat-W$\ddot{\mathrm{u}}$sthoff (GBW) and Bartels,
Golec-Biernat and Kowalski (BGK )models. The main elements are based on the
color dipole picture (CDP) and the generalized double asymptotic scaling (DAS)
approach for usual parton distribution functions (PDFs). The comparisons with
the HERA data are made and predictions for the proposed LHeC and FCC-he
colliders are also provided in a wide range of the transverse separation $r$.
In particular, the ratio $R^{h}=F_{L}^{h}/F_{2}^{h}, h=c,b,t$ is well described
by the dipole models and is sensitive to the collider energies from HERA until
FCC-he. We derive correlated bounds on the ratio $F^{c}_{2}/F_{2}$ and
$F^{b}_{2}/F_{2}$ and compared them with the BGK and IP-sat models. The
uncertainties are due to the renormalization and factorization scales at large
and low $r$ values. The Sudakov form factor into the heavy quark structure
functions is incorporated and the results are considered, which are dependent
on the hard scale in a wide range of the transverse separation $r$.
