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The Optimal Upper and Lower Power Mean Bounds for a Convex Combination of the Arithmetic and Logarithmic Means

Author

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  • Wei-Feng Xia
  • Yu-Ming Chu
  • Gen-Di Wang

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 ?

Suggested Citation

  • Wei-Feng Xia & Yu-Ming Chu & Gen-Di Wang, 2010. "The Optimal Upper and Lower Power Mean Bounds for a Convex Combination of the Arithmetic and Logarithmic Means," Abstract and Applied Analysis, Hindawi, vol. 2010, pages 1-9, April.
    • Handle: RePEc:hin:jnlaaa:604804
    DOI: 10.1155/2010/604804
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    Cited by:

    1. Wei-Mao Qian & Bo-Yong Long, 2012. "Sharp Bounds by the Generalized Logarithmic Mean for the Geometric Weighted Mean of the Geometric and Harmonic Means," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    2. Wei-Mao Qian & Zhong-Hua Shen, 2012. "Inequalities between Power Means and Convex Combinations of the Harmonic and Logarithmic Means," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    3. Hongya Gao & Jianling Guo & Wanguo Yu, 2011. "Sharp Bounds for Power Mean in Terms of Generalized Heronian Mean," Abstract and Applied Analysis, John Wiley & Sons, vol. 2011(1).
    4. Yong-Min Li & Bo-Yong Long & Yu-Ming Chu & Wei-Ming Gong, 2012. "Optimal Inequalities for Power Means," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    5. Yu-Ming Chu & Ye-Fang Qiu & Miao-Kun Wang, 2010. "Sharp Power Mean Bounds for the Combination of Seiffert and Geometric Means," Abstract and Applied Analysis, John Wiley & Sons, vol. 2010(1).
    6. Yong-Min Li & Bo-Yong Long & Yu-Ming Chu, 2012. "A Best Possible Double Inequality for Power Mean," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).

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