Copyright © 2010 Ovidiu Bagdasar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let n be a positive integer and p, q, a, and b real numbers satisfying p>q>0 and 0<a<b. It is proved that for the real numbers a1,…,an∈[a,b], the maximum of the function fp,q(a1,…,an)=(a1p+⋯+anp)/n-((a1q+⋯+anq)/n)p/q is attained if and only if k(n) of the numbers a1,…,an are equal to a and the other n-k(n) are equal to b, while k(n) is one of the values [(bq-Dp,qq(a,b))/(bq-aq)⋅n], [(bq-Dp,qq(a,b))/(bq-aq)⋅n]+1, where [⋅] denotes the integer part and Dp,q(a,b) represents the Stolarsky mean of a and b, of powers p and q. Some asymptotic results concerning k(n) are also discussed.