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: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Hindawi Limited
2012-01-01
|
| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2012/471096 |
| id |
doaj-4cdd6d18126f42d79fcb40c4ae1582f6 |
|---|---|
| record_format |
Article |
| spelling |
doaj-4cdd6d18126f42d79fcb40c4ae1582f62020-11-24T23:22:40ZengHindawi LimitedJournal 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 |
| collection |
DOAJ |
| language |
English |
| format |
Article |
| sources |
DOAJ |
| author |
Wei-Mao Qian Zhong-Hua Shen |
| spellingShingle |
Wei-Mao Qian Zhong-Hua Shen Inequalities between Power Means and Convex Combinations of the Harmonic and Logarithmic Means Journal of Applied Mathematics |
| author_facet |
Wei-Mao Qian Zhong-Hua Shen |
| author_sort |
Wei-Mao Qian |
| title |
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_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_sort |
inequalities between power means and convex combinations of the harmonic and logarithmic means |
| publisher |
Hindawi Limited |
| series |
Journal of Applied Mathematics |
| issn |
1110-757X 1687-0042 |
| publishDate |
2012-01-01 |
| 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. |
| url |
http://dx.doi.org/10.1155/2012/471096 |
| work_keys_str_mv |
AT weimaoqian inequalitiesbetweenpowermeansandconvexcombinationsoftheharmonicandlogarithmicmeans AT zhonghuashen inequalitiesbetweenpowermeansandconvexcombinationsoftheharmonicandlogarithmicmeans |
| _version_ |
1725566983629438976 |