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Abstract and Applied Analysis
Volume 2011, Article ID 679201, 9 pages
http://dx.doi.org/10.1155/2011/679201
Research Article

Sharp Bounds for Power Mean in Terms of Generalized Heronian Mean

College of Mathematics and Computer Science, Hebei University, Baoding 071002, China

Received 2 March 2011; Accepted 7 April 2011

Academic Editor: Marcia Federson

Copyright © 2011 Hongya Gao 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

For 1<𝑟<+, we find the least value 𝛼 and the greatest value 𝛽 such that the inequality 𝐻𝛼(𝑎,𝑏)<𝐴𝑟(𝑎,𝑏)<𝐻𝛽(𝑎,𝑏) holds for all 𝑎,𝑏>0 with 𝑎𝑏. Here, 𝐻𝜔(𝑎,𝑏) and 𝐴𝑟(𝑎,𝑏) are the generalized Heronian and the power means of two positive numbers 𝑎 and 𝑏, respectively.