Inequalities Involving the Mean and the Standard Deviation of Nonnegative Real Numbers

Letm(y)=∑j=1yj/n and s(y)= √ m(y2)−m2(y) be themean and the standard deviation of the components of the vector y = (y1, y2, . . . , yn−1, yn), where yq = (y 1 , y 2 , . . . , y n−1, y n) with q a positive integer. Here, we prove that if y ≥ 0, then m(y2p) + (1/√n− 1)s(y2p) ≤ √ m(y2 ) + (1/ √ n− 1)s(y2 ) for p = 0,1,2, . . . . The equality holds if and only if the (n − 1) largest components of y are equal. It follows that (l2p(y))p=0, l2p(y) = (m(y2 p ) + (1/ √ n− 1)s(y2p))2 , is a strictly increasing sequence converging to y1, the largest component of y, except if the (n− 1) largest components of y are equal. In this case, l2p(y)= y1 for all p.