A Sharp Double Inequality between Harmonic and Identric MeansReport as inadecuate

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Abstract and Applied AnalysisVolume 2011 2011, Article ID 657935, 7 pages

Research Article

Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China

Department of Mathematics, Hangzhou Normal University, Hangzhou 310012, China

Received 31 May 2011; Accepted 6 August 2011

Academic Editor: Ondřej Došlý

Copyright © 2011 Yu-Ming Chu 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.


We find the greatest value 𝑝 and the least value 𝑞 in 0,1-2 such that the double inequality 𝐻𝑝𝑎+1−𝑝𝑏,𝑝𝑏+1−𝑝𝑎<𝐼𝑎,𝑏<𝐻𝑞𝑎+1−𝑞𝑏,𝑞𝑏+1−𝑞𝑎 holds for all 𝑎,𝑏>0 with 𝑎≠𝑏. Here, 𝐻𝑎,𝑏, and 𝐼𝑎,𝑏 denote the harmonic and identric means of two positive numbers 𝑎 and 𝑏, respectively.

Author: Yu-Ming Chu, Miao-Kun Wang, and Zi-Kui Wang

Source: https://www.hindawi.com/


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