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 ...
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| Format: | Article |
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Wiley
2012-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2012/471096 |
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| author | Wei-Mao Qian Zhong-Hua Shen |
| author_facet | Wei-Mao Qian Zhong-Hua Shen |
| author_sort | Wei-Mao Qian |
| collection | DOAJ |
| description | 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 a−log b), and Mp(a,b)=((ap+bp)/2)1/p (p≠0)
and M0(a,b)=ab are the harmonic, logarithmic, and pth power means of a and b, respectively. |
| format | Article |
| id | doaj-art-4cdd6d18126f42d79fcb40c4ae1582f6 |
| institution | Kabale University |
| issn | 1110-757X 1687-0042 |
| language | English |
| publishDate | 2012-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Applied Mathematics |
| spelling | doaj-art-4cdd6d18126f42d79fcb40c4ae1582f62025-02-03T05:47:35ZengWileyJournal of Applied Mathematics1110-757X1687-00422012-01-01201210.1155/2012/471096471096Inequalities between Power Means and Convex Combinations of the Harmonic and Logarithmic MeansWei-Mao Qian0Zhong-Hua Shen1School of Distance Education, Huzhou Broadcast and TV University, Huzhou 313000, ChinaDepartment of Mathematics, Hangzhou Normal University, Hangzhou 310036, ChinaWe 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 a−log b), and Mp(a,b)=((ap+bp)/2)1/p (p≠0) and M0(a,b)=ab are the harmonic, logarithmic, and pth power means of a and b, respectively.http://dx.doi.org/10.1155/2012/471096 |
| spellingShingle | Wei-Mao Qian Zhong-Hua Shen Inequalities between Power Means and Convex Combinations of the Harmonic and Logarithmic Means Journal of Applied Mathematics |
| title | Inequalities between Power Means and Convex Combinations of the Harmonic and Logarithmic Means |
| title_full | Inequalities between Power Means and Convex Combinations of the Harmonic and Logarithmic Means |
| title_fullStr | Inequalities between Power Means and Convex Combinations of the Harmonic and Logarithmic Means |
| title_full_unstemmed | Inequalities between Power Means and Convex Combinations of the Harmonic and Logarithmic Means |
| title_short | Inequalities between Power Means and Convex Combinations of the Harmonic and Logarithmic Means |
| title_sort | inequalities between power means and convex combinations of the harmonic and logarithmic means |
| url | http://dx.doi.org/10.1155/2012/471096 |
| work_keys_str_mv | AT weimaoqian inequalitiesbetweenpowermeansandconvexcombinationsoftheharmonicandlogarithmicmeans AT zhonghuashen inequalitiesbetweenpowermeansandconvexcombinationsoftheharmonicandlogarithmicmeans |