We prove that for all integers n ≥ 2 the expression B_n-1 B_n+1 - B_n² can be represented as an infinite series with nonnegative terms. Here B_k denotes the k-th Bell number. It follows that the sequence (B_n)_{n ≥ 0} is strictly log-convex. This result refines Engel's inequality B_n² ≤ B_n-1 B_n+1.