A Sharp Double Inequality between Seiffert, Arithmetic, and Geometric Means

Wei-Ming Gong, Ying-Qing Song, Miao-Kun Wang and Yu-Ming Chu

Abstract and Applied Analysis, 2012, vol. 2012, 1-7

Abstract:

For fixed ð ‘ â‰¥ 1 and any ð ‘¡ 1 , ð ‘¡ 2 ∈ ( 0 , 1 / 2 ) we prove that the double inequality ð º ð ‘ ( ð ‘¡ 1 ð ‘Ž + ( 1 − ð ‘¡ 1 ) ð ‘ , ð ‘¡ 1 ð ‘ + ( 1 − ð ‘¡ 1 ) ð ‘Ž ) ð ´ 1 − ð ‘ ( ð ‘Ž , ð ‘ ) < 𠑃 ( ð ‘Ž , ð ‘ ) < ð º ð ‘ ( ð ‘¡ 2 ð ‘Ž + ( 1 − ð ‘¡ 2 ) ð ‘ , ð ‘¡ 2 ð ‘ + ( 1 − ð ‘¡ 2 ) ð ‘Ž ) ð ´ 1 − ð ‘ ( ð ‘Ž , ð ‘ ) holds for all ð ‘Ž , ð ‘ > 0 with ð ‘Ž â‰ ð ‘ if and only if ð ‘¡ 1 √ ≤ ( 1 − 1 − ( 2 / 𠜋 ) 2 / ð ‘ ) / 2 and ð ‘¡ 2 √ ≥ ( 1 − 1 / 3 ð ‘ ) / 2 . Here, 𠑃 ( ð ‘Ž , ð ‘ ) , ð ´ ( ð ‘Ž , ð ‘ ) and ð º ( ð ‘Ž , ð ‘ ) denote the Seiffert, arithmetic, and geometric means of two positive numbers ð ‘Ž and ð ‘ , respectively.

Date: 2012
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnlaaa:684834

DOI: 10.1155/2012/684834

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