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An Approximation Formula for Nielsen’s Beta Function Involving the Trigamma Function

Author

Listed:
  • Mansour Mahmoud

    (Mathematics Department, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia)

  • Hanan Almuashi

    (Mathematics Department, Faculty of Science, Jeddah University, P.O. Box 80327, Jeddah 21589, Saudi Arabia)

Abstract

We prove that the function σ ( s ) defined by β ( s ) = 6 s 2 + 12 s + 5 3 s 2 ( 2 s + 3 ) − ψ ′ ( s ) 2 − σ ( s ) 2 s 5 , s > 0 , is strictly increasing with the sharp bounds 0 < σ ( s ) < 49 120 , where β ( s ) is Nielsen’s beta function and ψ ′ ( s ) is the trigamma function. Furthermore, we prove that the two functions s ↦ ( − 1 ) 1 + μ β ( s ) − 6 s 2 + 12 s + 5 3 s 2 ( 2 s + 3 ) + ψ ′ ( s ) 2 + 49 μ 240 s 5 , μ = 0 , 1 are completely monotonic for s > 0 . As an application, double inequality for β ( s ) involving ψ ′ ( s ) is obtained, which improve some recent results.

Suggested Citation

  • Mansour Mahmoud & Hanan Almuashi, 2022. "An Approximation Formula for Nielsen’s Beta Function Involving the Trigamma Function," Mathematics, MDPI, vol. 10(24), pages 1-8, December.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:24:p:4729-:d:1001734
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    References listed on IDEAS

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    1. Christian Berg & Stamatis Koumandos & Henrik L. Pedersen, 2021. "Nielsen's beta function and some infinitely divisible distributions," Mathematische Nachrichten, Wiley Blackwell, vol. 294(3), pages 426-449, March.
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