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...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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Hindawi Limited
2010-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2010/604804 |
| Summary: | 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? |
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| ISSN: | 1085-3375 1687-0409 |