Inequalities involving the mean and the standard of nonnegative real numbers

Abstract

Letm(y)= ∑ j=1 n yj /n and s(y)= m(y2)-m2 (y) be the mean and the standard deviation of the components of the vector y=(y1,y2,...,y n-1,yn), where yq =(y1q,y2q,...,y n-1 q,ynq) with q a positive integer. Here, we prove that if y<0,then m(y 2p)+(1/ n-1)s(y 2p)≤ m(y 2p +1)+(1/ n-1)s(y 2p +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 ∞, l 2p (y)= (m(y 2p)+(1/ n-1)s(y 2p)) 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, l 2p (y)=y1 for all p.