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A Sharp Double Inequality between Harmonic and Identric Means

Author

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

Abstract

We find the greatest value ð ‘ and the least value ð ‘ž in ( 0 , 1 / 2 ) such that the double inequality ð » ( ð ‘ ð ‘Ž + ( 1 − ð ‘ ) ð ‘ , ð ‘ ð ‘ + ( 1 − ð ‘ ) ð ‘Ž ) < ð ¼ ( ð ‘Ž , ð ‘ ) < ð » ( ð ‘ž ð ‘Ž + ( 1 − ð ‘ž ) ð ‘ , ð ‘ž ð ‘ + ( 1 − ð ‘ž ) ð ‘Ž ) holds for all ð ‘Ž , ð ‘ > 0 with ð ‘Ž â‰ ð ‘ . Here, ð » ( ð ‘Ž , ð ‘ ) , and ð ¼ ( ð ‘Ž , ð ‘ ) denote the harmonic and identric means of two positive numbers ð ‘Ž and ð ‘ , respectively.

Suggested Citation

  • Yu-Ming Chu & Miao-Kun Wang & Zi-Kui Wang, 2011. "A Sharp Double Inequality between Harmonic and Identric Means," Abstract and Applied Analysis, Hindawi, vol. 2011, pages 1-7, October.
    • Handle: RePEc:hin:jnlaaa:657935
    DOI: 10.1155/2011/657935
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    Cited by:

    1. Ling Zhu, 2021. "New Bounds for the Sine Function and Tangent Function," Mathematics, MDPI, vol. 9(19), pages 1-12, September.
    2. Tareq Saeed & Muhammad Bilal Khan & Savin Treanță & Hamed H. Alsulami & Mohammed Sh. Alhodaly, 2023. "Study of Log Convex Mappings in Fuzzy Aunnam Calculus via Fuzzy Inclusion Relation over Fuzzy-Number Space," Mathematics, MDPI, vol. 11(9), pages 1-16, April.
    3. Ling Zhu, 2021. "New Inequalities of Cusa–Huygens Type," Mathematics, MDPI, vol. 9(17), pages 1-13, August.
    4. Xuexiao You & Muhammad Adil Khan & Hidayat Ullah & Tareq Saeed, 2022. "Improvements of Slater’s Inequality by Means of 4-Convexity and Its Applications," Mathematics, MDPI, vol. 10(8), pages 1-19, April.
    5. Hidayat Ullah & Muhammad Adil Khan & Tareq Saeed, 2021. "Determination of Bounds for the Jensen Gap and Its Applications," Mathematics, MDPI, vol. 9(23), pages 1-29, December.

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