the optimal upper and lower power mean bounds for a convex combination of the arithmetic and logarithmic means
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2010
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Abstract
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?
| Reference Key |
xia2010abstractthe
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|---|---|
| Authors | ;Wei-Feng Xia;Yu-Ming Chu;Gen-Di Wang |
| Journal | science and technology of advanced materials |
| Year | 2010 |
| DOI |
10.1155/2010/604804
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| URL | |
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