Optimal Inequalities between Harmonic, Geometric, Logarithmic, and Arithmetic-Geometric Means

Yu-Ming Chu and Miao-Kun Wang

Journal of Applied Mathematics, 2011, vol. 2011, 1-9

Abstract:

We find the least values ð ‘ , ð ‘ž , and ð ‘ in (0, 1/2) such that the inequalities ð » ( ð ‘ ð ‘Ž + ( 1 − ð ‘ ) ð ‘ , ð ‘ ð ‘ + ( 1 − ð ‘ ) ð ‘Ž ) > A G ( ð ‘Ž , ð ‘ ) , ð º ( ð ‘ž ð ‘Ž + ( 1 − ð ‘ž ) ð ‘ , ð ‘ž ð ‘ + ( 1 − ð ‘ž ) ð ‘Ž ) > A G ( ð ‘Ž , ð ‘ ) , and ð ¿ ( ð ‘ ð ‘Ž + ( 1 − ð ‘ ) ð ‘ , ð ‘ ð ‘ + ( 1 − ð ‘ ) ð ‘Ž ) > A G ( ð ‘Ž , ð ‘ ) hold for all ð ‘Ž , ð ‘ > 0 with ð ‘Ž â‰ ð ‘ , respectively. Here A G ( ð ‘Ž , ð ‘ ) , ð » ( ð ‘Ž , ð ‘ ) , ð º ( ð ‘Ž , ð ‘ ) , and ð ¿ ( ð ‘Ž , ð ‘ ) denote the arithmetic-geometric, harmonic, geometric, and logarithmic means of two positive numbers ð ‘Ž and ð ‘ , respectively.

Date: 2011
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnljam:618929

DOI: 10.1155/2011/618929

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