  We review the main results on the generalization of the DGLAP evolution
equations within the cut Mellin moments (CMM) approach, which allows one to
overcome the problem of kinematic constraints in Bjorken $x$. CMM obtained by
multiple integrations as well as multiple differentiations of the original
parton distribution also satisfy the DGLAP equations with the simply
transformed evolution kernel. The CMM approach provides novel tools to test
QCD; here we present one of them. Using appropriate classes of CMM, we
construct the generalized Bjorken sum rule that allows us to determine the
Bjorken sum rule value from the experimental data in a restricted kinematic
range of $x$. We apply our analysis to COMPASS data on the spin structure
function $g_1$.
