Abstract and Applied Analysis
Volume 2009 (2009), Article ID 694394, 10 pages
doi:10.1155/2009/694394
Research Article

Optimal Inequalities among Various Means of Two Arguments

Ming-yu Shi,1 Yu-ming Chu,2 and Yue-ping Jiang1

1College of Mathematics and Econometrics, Hunan University, Changsha 410082, China
2Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China

Received 10 September 2009; Accepted 19 October 2009

Academic Editor: Dumitru Baleanu

Copyright © 2009 Ming-yu Shi et al. 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

We establish two optimal inequalities among power mean Mp(a,b)=(ap/2+bp/2)1/p, arithmetic mean A(a,b)=(a+b)/2, logarithmic mean L(a,b)=(a−b)/(log⁡a−log⁡b), and geometric mean G(a,b)=ab.