Let k be a non-negative integer, and let r 1 ,r 2 r k be non-negative real numbers satisfying r 1 + r 2 + + r k 1 and r i + 1 + + r k < (k - i)/k for all i = 1 k - 1. It is proved that there exists a constant c such that for any X 1 ,X 2 X k non-negative i.i.d. random variables, if X ( j ) denotes the jth order statistic, then the following inequality holds: EX r 1 ( 1 ) X r k ( k ) cEX ( 1 ) . Moreover, it is shown that the conditions on r 1 r k are best possible for the inequality to hold.