  In this paper, we consider the generalized fractional maximal function and
use it to introduce the space of generalized fractional maximal functions and
the various cones of monotone functions generated by generalized fractional
maximal functions $M_\Phi f$. We introduced three function classes. We give
equivalent descriptions of such cones when the function $\Phi$ belongs to some
function classes. The conditions for their mutual covering are given. Then,
these cones are used to construct a criterion for embedding the space of
generalized fractional maximal functions into the rearrangement invariant
spaces (RIS). The optimal RIS for such embedding is also described.
