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

The ratio of the beauty structure functions $R^b = F^b_L/F^b_2$ at low x