Abstract
Let
m(y)=∑j=1nyj/n
and s(y)=m(y2)−m2(y)
be the mean and the standard deviation of the components of the vector y=(y1,y2,…,yn−1,yn), where yq= (y1q,y2q,…,yn−1q,ynq) with q a positive integer. Here, we prove that if y≥0,
then m(y2p)+(1/n−1)s(y2p)≤m(y2p+1)+(1/n−1)s(y2p+1)
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(y2p)+(1/n−1)s(y2p))2−p,
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.