Inequalities between Power Means and Convex Combinations of the Harmonic and Logarithmic Means

We prove that αH(a,b)+(1−α)L(a,b)>M(1−4α)/3(a,b) for α∈(0,1) and all a,b>0 with a≠b if and only if α∈[1/4,1) and αH(a,b)+(1−α)L(a,b)

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Bibliographic Details
Published in:Journal of Applied Mathematics 2012-01, Vol.2012 (2012), p.594-607-342
Main Authors: Qian, Wei-Mao, Shen, Zhong-Hua
Format: Article
Language:English
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Summary:We prove that αH(a,b)+(1−α)L(a,b)>M(1−4α)/3(a,b) for α∈(0,1) and all a,b>0 with a≠b if and only if α∈[1/4,1) and αH(a,b)+(1−α)L(a,b)
ISSN:1110-757X
1687-0042
DOI:10.1155/2012/471096