  We generalize a result of Garvan on inequalities and interpretations of the
moments of the partition rank and crank functions. In particular for nearly 30
Bailey pairs, we introduce a rank-like function, establish inequalities with
the moments of the rank-like function and an associated crank-like function,
and give an associated so called higher order smallest parts function. In some
cases we are able to deduce inequalities among the rank-like functions. We also
conjecture additional inequalities and a large number of congruences for the
higher order smallest parts functions.
