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An Optimal Double Inequality between Seiffert and Geometric Means

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  • Yu-Ming Chu
  • Miao-Kun Wang
  • Zi-Kui Wang

Abstract

For ð ›¼ , ð ›½ ∈ ( 0 , 1 / 2 ) we prove that the double inequality ð º ( ð ›¼ ð ‘Ž + ( 1 − ð ›¼ ) ð ‘ , ð ›¼ ð ‘ + ( 1 − ð ›¼ ) ð ‘Ž ) < 𠑃 ( ð ‘Ž , ð ‘ ) < ð º ( ð ›½ ð ‘Ž + ( 1 − ð ›½ ) ð ‘ , ð ›½ ð ‘ + ( 1 − ð ›½ ) ð ‘Ž ) holds for all ð ‘Ž , ð ‘ > 0 with ð ‘Ž â‰ ð ‘ if and only if √ ð ›¼ ≤ ( 1 − 1 − 4 / 𠜋 2 ) / 2 and √ ð ›½ ≥ ( 3 − 3 ) / 6 . Here, ð º ( ð ‘Ž , ð ‘ ) and 𠑃 ( ð ‘Ž , ð ‘ ) denote the geometric and Seiffert means of two positive numbers ð ‘Ž and ð ‘ , respectively.

Suggested Citation

  • Yu-Ming Chu & Miao-Kun Wang & Zi-Kui Wang, 2011. "An Optimal Double Inequality between Seiffert and Geometric Means," Journal of Applied Mathematics, Hindawi, vol. 2011, pages 1-6, November.
    • Handle: RePEc:hin:jnljam:261237
    DOI: 10.1155/2011/261237
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