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

Mansour Mahmoud () and Hanan Almuashi
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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

Mathematics, 2022, vol. 10, issue 24, 1-8

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.

Keywords: Nielsen’s beta function; trigamma function; approximation formula; completely monotonic; sharp bound (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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