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

Abstract and Applied Analysis, 2010, vol. 2010, 1-9

Abstract:

For ð ‘ âˆˆ â„ , the power mean ð ‘€ ð ‘ ( ð ‘Ž , ð ‘ ) of order ð ‘ , logarithmic mean ð ¿ ( ð ‘Ž , ð ‘ ) , and arithmetic mean ð ´ ( ð ‘Ž , ð ‘ ) of two positive real values ð ‘Ž and ð ‘ are defined by ð ‘€ ð ‘ ( ð ‘Ž , ð ‘ ) = ( ( ð ‘Ž ð ‘ + ð ‘ ð ‘ ) / 2 ) 1 / ð ‘ , for ð ‘ â‰ 0 and ð ‘€ ð ‘ âˆš ( ð ‘Ž , ð ‘ ) = ð ‘Ž ð ‘ , for ð ‘ = 0 , ð ¿ ( ð ‘Ž , ð ‘ ) = ( ð ‘ âˆ’ ð ‘Ž ) / ( l o g ð ‘ âˆ’ l o g ð ‘Ž ) , for ð ‘Ž â‰ ð ‘ and ð ¿ ( ð ‘Ž , ð ‘ ) = ð ‘Ž , for ð ‘Ž = ð ‘ and ð ´ ( ð ‘Ž , ð ‘ ) = ( ð ‘Ž + ð ‘ ) / 2 , respectively. In this paper, we answer the question: for ð ›¼ ∈ ( 0 , 1 ) , what are the greatest value ð ‘ and the least value ð ‘ž , such that the double inequality ð ‘€ ð ‘ ( ð ‘Ž , ð ‘ ) ≤ ð ›¼ ð ´ ( ð ‘Ž , ð ‘ ) + ( 1 − ð ›¼ ) ð ¿ ( ð ‘Ž , ð ‘ ) ≤ ð ‘€ ð ‘ž ( ð ‘Ž , ð ‘ ) holds for all ð ‘Ž , ð ‘ > 0 ?

Date: 2010
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnlaaa:604804

DOI: 10.1155/2010/604804

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