An inequality for order statistics

Let k be a non-negative integer, and let r1, r2,..., rk be non-negative real numbers satisfying r1 + r2 + ... + rk [less-than-or-equals, slant] 1 and ri+1 + ... + rk < (k - i)/k for all i = 1,...,k - 1. It is proved that there exists a constant c such that for any X1,X2,...,Xk non-negative i.i.d. random variables, if X(j) denotes the jth order statistic, then the following inequality holds: EXr1(1)... Xrkk[less-than-or-equals, slant]cEX(1). Moreover, it is shown that the conditions on r1,...,rk are best possible for the inequality to hold.