A prophet inequality for independent random variables with finite variances

It is demonstrated that for each n greater than or equal to 2 there exists a minimal universal constant, c(n), such that, for any sequence of independent random variables {X-r, r greater than or equal to 1} with finite variances, E [max(1 less than or equal to i less than or equal to n)X(i)]-sup(T)EX(T) less than or equal to c(n) root n-1max(1 less than or equal to i less than or equal to n) root Var(X-i), where the supremum is over all stopping times T, 1 less than or equal to T less than or equal to n. Furthermore, c(n) less than or equal to 1/2 and lim inf(n-->infinity) c(n) greater than or equal to 0.439485....