Abstract and Applied Analysis

The Optimal Upper and Lower Power Mean Bounds for a Convex Combination of the Arithmetic and Logarithmic Means

Wei-Feng Xia, Yu-Ming Chu, and Gen-Di Wang

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Abstract

For p , the power mean M p ( 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 M p ( a , b ) = ( ( a p + b p ) / 2 ) 1 / p , for p 0 and M p ( a , b ) = a b , for p = 0 , L ( a , b ) = ( b - a ) / ( log b - log a ) , 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 M p ( a , b ) α A ( a , b ) + ( 1 - α ) L ( a , b ) M q ( a , b ) holds for all a , b > 0 .

Article information

Source
Abstr. Appl. Anal., Volume 2010 (2010), Article ID 604804, 9 pages.

Dates
First available in Project Euclid: 1 November 2010

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1288620716

Digital Object Identifier
doi:10.1155/2010/604804

Mathematical Reviews number (MathSciNet)
MR2629623

Zentralblatt MATH identifier
1190.26038

Citation

Xia, Wei-Feng; Chu, Yu-Ming; Wang, Gen-Di. The Optimal Upper and Lower Power Mean Bounds for a Convex Combination of the Arithmetic and Logarithmic Means. Abstr. Appl. Anal. 2010 (2010), Article ID 604804, 9 pages. doi:10.1155/2010/604804. https://projecteuclid.org/euclid.aaa/1288620716


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