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Abstract and Applied Analysis
Volume 2011, Article ID 657935, 7 pages
Research Article

A Sharp Double Inequality between Harmonic and Identric Means

1Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China
2Department of Mathematics, Hangzhou Normal University, Hangzhou 310012, China

Received 31 May 2011; Accepted 6 August 2011

Academic Editor: OndΕ™ej DoΕ‘lΓ½

Copyright Β© 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 π‘ž in (0,1/2) such that the double inequality 𝐻(π‘π‘Ž+(1βˆ’π‘)𝑏,𝑝𝑏+(1βˆ’π‘)π‘Ž)<𝐼(π‘Ž,𝑏)<𝐻(π‘žπ‘Ž+(1βˆ’π‘ž)𝑏,π‘žπ‘+(1βˆ’π‘ž)π‘Ž) holds for all π‘Ž,𝑏>0 with π‘Žβ‰ π‘. Here, 𝐻(π‘Ž,𝑏), and 𝐼(π‘Ž,𝑏) denote the harmonic and identric means of two positive numbers π‘Ž and 𝑏, respectively.