The Moment Bound is Tighter than Chernoff's Bound for Positive Tail Probabilities
Thomas K. Philips, Randolph D. Nelson · The American Statistician · 1995
Chernoff's bound on P[X ⩾ t] is used almost universally when a tight bound on tail probabilities is required. In this article we show that for all positive t and for all distributions, the moment bound is tighter than Chernoff's bound. By way of example, we demonstrate that the improvement is often substantial.