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)<M(1−4α)/3(a,b) if and only if α∈(0,3345/80−11/16), and the parameter (1−4α)/3 is the best possible in either case. Here, H(a,b)=2ab/(a+b), L(a,b)=(a−b)/(log ...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Hindawi Limited
2012-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2012/471096 |