Engelʹs inequality for bell numbers

K. Engel has conjectured that the average number of blocks in a partition of an n-set is a concave function of n. The average in question is a quotient of two Bell numbers less 1, and we prove Engelʹs conjecture for all n sufficiently large by an extension of the Moser-Wyman asymptotic formula for the Bell numbers. We also give a general theorem which specializes to an inequality about Bell numbers less complex than Engelʹs, in the fewer terms of the asymptotic expansion are needed to verify it for all sufficiently large n.