A sharp inequality for martingales with values in \ell_\infty^n

A classical result of Burkholder asserts that a Banach space is UMD if and only if it is zeta-convex. There is a dual characterization due to Lee: a Banach space is UMD if and only if there is a biconcave function satisfying appropriate majorization. We will be mainly concerned with the space \ell_\infty^n. Our purpose will be to establish an estimate strictly related to the dual characterization, which will yield a sharp inequality for martingale transforms.