Urn Sampling and a Majorization Inequality

In the context of a certain urn-sampling game, Bennett has studied pairs of sequences for which the products of successive finite differences of the sequences are majorized by the differences of the termwise product of the sequences. Bennett conjectured that the sequences {\bf x}_n={A-n \choose a} and {\bf y}_n={B-n \choose b} form such a pair for any nonnegative integersA⩾a,B⩾b, and proved this result in the cases min{A, B}⩾a+band −b⩽A−B⩽a. We complete the proof of the conjecture by proving the result under the assumption max{A, B}⩾a+b.