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Inequalities between Power Means and Convex Combinations of the Harmonic and Logarithmic Means

Author

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  • Wei-Mao Qian
  • Zhong-Hua Shen

Abstract

We prove that ð ›¼ ð » ( ð ‘Ž , ð ‘ ) + ( 1 − ð ›¼ ) ð ¿ ( ð ‘Ž , ð ‘ ) > ð ‘€ ( 1 − 4 ð ›¼ ) / 3 ( ð ‘Ž , ð ‘ ) for ð ›¼ ∈ ( 0 , 1 ) and all ð ‘Ž , ð ‘ > 0 with ð ‘Ž â‰ ð ‘ if and only if ð ›¼ ∈ [ 1 / 4 , 1 ) and ð ›¼ ð » ( ð ‘Ž , ð ‘ ) + ( 1 − ð ›¼ ) ð ¿ ( ð ‘Ž , ð ‘ ) < ð ‘€ ( 1 − 4 ð ›¼ ) / 3 ( ð ‘Ž , ð ‘ ) if and only if √ ð ›¼ ∈ ( 0 , 3 3 4 5 / 8 0 − 1 1 / 1 6 ) , and the parameter ( 1 − 4 ð ›¼ ) / 3 is the best possible in either case. Here, ð » ( ð ‘Ž , ð ‘ ) = 2 ð ‘Ž ð ‘ / ( ð ‘Ž + ð ‘ ) , ð ¿ ( ð ‘Ž , ð ‘ ) = ( ð ‘Ž − ð ‘ ) / ( l o g ð ‘Ž − l o g ð ‘ ) , and ð ‘€ ð ‘ ( ð ‘Ž , ð ‘ ) = ( ( ð ‘Ž ð ‘ + ð ‘ ð ‘ ) / 2 ) 1 / ð ‘ ( ð ‘ â‰ 0 ) and ð ‘€ 0 √ ( ð ‘Ž , ð ‘ ) = ð ‘Ž ð ‘ are the harmonic, logarithmic, and p th power means of a and b , respectively.

Suggested Citation

  • Wei-Mao Qian & Zhong-Hua Shen, 2012. "Inequalities between Power Means and Convex Combinations of the Harmonic and Logarithmic Means," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-14, February.
    • Handle: RePEc:hin:jnljam:471096
    DOI: 10.1155/2012/471096
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