Journal of Inequalities and Applications
Volume 2010 (2010), Article ID 492570, 10 pages
Research Article

Complementary Inequalities Involving the Stolarsky Mean

Department of Mathematical Sciences, The University of Nottingham, University Park, Nottingham NG7 2RD, UK

Received 24 February 2010; Accepted 1 May 2010

Academic Editor: László Losonczi

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.


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.