The Optimal Upper and Lower Power Mean Bounds for a Convex Combination of the Arithmetic and Logarithmic Means
For p∈ℝ, the power mean Mp(a,b) of order p, logarithmic mean L(a,b), and arithmetic mean A(a,b) of two positive real values a and b are defined by Mp(a,b)=((ap+bp)/2)1/p, for p≠0 and Mp(a,b)=ab, for p=0, L(a,b)=(b-a)/(logb-loga), for a≠b and L(a,b)=a, for a=b and A(a,b)=(a+b)/2, respectively. In t...
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Wiley
2010-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2010/604804 |
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| author | Wei-Feng Xia Yu-Ming Chu Gen-Di Wang |
| author_facet | Wei-Feng Xia Yu-Ming Chu Gen-Di Wang |
| author_sort | Wei-Feng Xia |
| collection | DOAJ |
| description | For p∈ℝ, the power mean Mp(a,b) of order p, logarithmic mean L(a,b), and arithmetic mean A(a,b) of two positive real values a and b are defined by Mp(a,b)=((ap+bp)/2)1/p, for p≠0 and Mp(a,b)=ab, for p=0, L(a,b)=(b-a)/(logb-loga), for a≠b and L(a,b)=a, for a=b and A(a,b)=(a+b)/2, respectively. In this paper, we answer the question: for α∈(0,1), what are the greatest value p and the least value q, such that the double inequality Mp(a,b)≤αA(a,b)+(1-α)L(a,b)≤Mq(a,b) holds for all a,b>0? |
| format | Article |
| id | doaj-art-3812b5f08ee54703a2523d23dd7ecde7 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2010-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-3812b5f08ee54703a2523d23dd7ecde72025-02-03T06:05:19ZengWileyAbstract and Applied Analysis1085-33751687-04092010-01-01201010.1155/2010/604804604804The Optimal Upper and Lower Power Mean Bounds for a Convex Combination of the Arithmetic and Logarithmic MeansWei-Feng Xia0Yu-Ming Chu1Gen-Di Wang2School of Teacher Education, Huzhou Teachers College, Huzhou, Zhejiang 313000, ChinaDepartment of Mathematics, Huzhou Teachers College, Huzhou, Zhejiang 313000, ChinaDepartment of Mathematics, Huzhou Teachers College, Huzhou, Zhejiang 313000, ChinaFor p∈ℝ, the power mean Mp(a,b) of order p, logarithmic mean L(a,b), and arithmetic mean A(a,b) of two positive real values a and b are defined by Mp(a,b)=((ap+bp)/2)1/p, for p≠0 and Mp(a,b)=ab, for p=0, L(a,b)=(b-a)/(logb-loga), for a≠b and L(a,b)=a, for a=b and A(a,b)=(a+b)/2, respectively. In this paper, we answer the question: for α∈(0,1), what are the greatest value p and the least value q, such that the double inequality Mp(a,b)≤αA(a,b)+(1-α)L(a,b)≤Mq(a,b) holds for all a,b>0?http://dx.doi.org/10.1155/2010/604804 |
| spellingShingle | Wei-Feng Xia Yu-Ming Chu Gen-Di Wang The Optimal Upper and Lower Power Mean Bounds for a Convex Combination of the Arithmetic and Logarithmic Means Abstract and Applied Analysis |
| title | The Optimal Upper and Lower Power Mean Bounds for a Convex Combination of the Arithmetic and Logarithmic Means |
| title_full | The Optimal Upper and Lower Power Mean Bounds for a Convex Combination of the Arithmetic and Logarithmic Means |
| title_fullStr | The Optimal Upper and Lower Power Mean Bounds for a Convex Combination of the Arithmetic and Logarithmic Means |
| title_full_unstemmed | The Optimal Upper and Lower Power Mean Bounds for a Convex Combination of the Arithmetic and Logarithmic Means |
| title_short | The Optimal Upper and Lower Power Mean Bounds for a Convex Combination of the Arithmetic and Logarithmic Means |
| title_sort | optimal upper and lower power mean bounds for a convex combination of the arithmetic and logarithmic means |
| url | http://dx.doi.org/10.1155/2010/604804 |
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