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Logarithmically Complete Monotonicity Properties Relating to the Gamma Function

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  • Tie-Hong Zhao
  • Yu-Ming Chu
  • Hua Wang

Abstract

We prove that the function 𠑓 𠛼 , 𠛽 ( 𠑥 ) = Γ 𠛽 ( 𠑥 + 𠛼 ) / 𠑥 𠛼 Γ ( 𠛽 𠑥 ) is strictly logarithmically completely monotonic on ( 0 , ∞ ) if √ ( 𠛼 , 𠛽 ) ∈ { ( 𠛼 , 𠛽 ) ∶ 1 / 𠛼 ≤ 𠛽 ≤ 1 , 𠛼 ≠1 } ∪ { ( 𠛼 , 𠛽 ) ∶ 0 < 𠛽 ≤ 1 , 𠜑 1 ( 𠛼 , 𠛽 ) ≥ 0 , 𠜑 2 ( 𠛼 , 𠛽 ) ≥ 0 } and [ 𠑓 𠛼 , 𠛽 ( 𠑥 ) ] − 1 is strictly logarithmically completely monotonic on ( 0 , ∞ ) if √ ( 𠛼 , 𠛽 ) ∈ { ( 𠛼 , 𠛽 ) ∶ 0 < 𠛼 ≤ 1 / 2 , 0 < 𠛽 ≤ 1 } ∪ { ( 𠛼 , 𠛽 ) ∶ 1 ≤ 𠛽 ≤ 1 / √ 𠛼 ≤ 2 , 𠛼 ≠1 } ∪ { ( 𠛼 , 𠛽 ) ∶ 1 / 2 ≤ 𠛼 < 1 , 𠛽 ≥ 1 / ( 1 − 𠛼 ) } , where 𠜑 1 ( 𠛼 , 𠛽 ) = ( 𠛼 2 + 𠛼 − 1 ) 𠛽 2 + ( 2 𠛼 2 − 3 𠛼 + 1 ) 𠛽 − 𠛼 and 𠜑 2 ( 𠛼 , 𠛽 ) = ( 𠛼 − 1 ) 𠛽 2 + ( 2 𠛼 2 − 5 𠛼 + 2 ) 𠛽 − 1 .

Suggested Citation

  • Tie-Hong Zhao & Yu-Ming Chu & Hua Wang, 2011. "Logarithmically Complete Monotonicity Properties Relating to the Gamma Function," Abstract and Applied Analysis, Hindawi, vol. 2011, pages 1-13, July.
    • Handle: RePEc:hin:jnlaaa:896483
    DOI: 10.1155/2011/896483
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    Cited by:

    1. Yang, Zhen-Hang & Chu, Yu-Ming & Zhang, Wen, 2019. "High accuracy asymptotic bounds for the complete elliptic integral of the second kind," Applied Mathematics and Computation, Elsevier, vol. 348(C), pages 552-564.
    2. Zhong-Xuan Mao & Ya-Ru Zhu & Bao-Hua Guo & Fu-Hai Wang & Yu-Hua Yang & Hai-Qing Zhao, 2021. "Qi Type Diamond-Alpha Integral Inequalities," Mathematics, MDPI, vol. 9(4), pages 1-24, February.
    3. Muhammad Bilal Khan & Eze R. Nwaeze & Cheng-Chi Lee & Hatim Ghazi Zaini & Der-Chyuan Lou & Khalil Hadi Hakami, 2023. "Weighted Fractional Hermite–Hadamard Integral Inequalities for up and down Ԓ-Convex Fuzzy Mappings over Coordinates," Mathematics, MDPI, vol. 11(24), pages 1-27, December.
    4. Muhammad Bilal Khan & Aleksandr Rakhmangulov & Najla Aloraini & Muhammad Aslam Noor & Mohamed S. Soliman, 2023. "Generalized Harmonically Convex Fuzzy-Number-Valued Mappings and Fuzzy Riemann–Liouville Fractional Integral Inequalities," Mathematics, MDPI, vol. 11(3), pages 1-24, January.

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