  We study the structure functions $F_{k}^{b}(x,Q^{2})$ ($k=2, L$) and the
reduced cross section $\sigma_{r}^{b}(x,Q^{2})$ for small values of
Bjorken$^{,}$s $x$ variable with respect to the hard (Lipatov) pomeron for the
gluon distribution and provide a compact formula for the ratio $R^{b}$ that is
useful to extract the beauty structure function from the beauty reduced cross
section, in particular at DESY HERA. Also we show that the effects of the
nonlinear corrections to the gluon distribution tame the behavior of the beauty
structure function and the beauty reduced cross section at low $x$.
