On Engel's Inequality for Bell Numbers

We prove that for all integers n ≥ 2 the expression Bn−1Bn+1 − B 2 n can be represented as an infinite series with nonnegative terms. Here Bk denotes the k-th Bell number. It follows that the sequence (Bn)n≥0 is strictly log-convex. This result refines Engel’s inequality B n ≤ Bn−1Bn+1.