Analytic properties and numerical representations for constructing the extended beta function using logarithmic mean

Mohammed Z. Alqarni; Mohamed Abdalla; Mohammed Z. Alqarni; Mohamed Abdalla

doi:10.3934/math.2024590

AIMS Mathematics

2024, Volume 9, Issue 5: 12072-12089. doi: 10.3934/math.2024590

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Analytic properties and numerical representations for constructing the extended beta function using logarithmic mean

Mohammed Z. Alqarni ^{1
,
,},
Mohamed Abdalla ²

1.
Mathematics Department, Faculty of Science, King Khalid University, Abha 61471, Saudi Arabia
2.
Mathematics Department, Faculty of Science, South Valley University, Qena 83523, Egypt

Received: 30 December 2023 Revised: 09 February 2024 Accepted: 29 February 2024 Published: 27 March 2024
MSC : 33B15, 33B99, 33C90

This paper aimed to obtain generalizations of both the logarithmic mean ($ \text{L}_{mean} $) and the Euler's beta function (EBF), which we call the extended logarithmic mean ($ \text{EL}_{mean} $) and the extended Euler's beta-logarithmic function (EEBLF), respectively. Also, we discussed various properties, including functional relations, inequalities, infinite sums, finite sums, integral formulas, and partial derivative representations, along with the Mellin transform for the EEBLF. Furthermore, we gave numerical comparisons between these generalizations and the previous studies using MATLAB R2018a in the form of tables and graphs. Additionally, we presented a new version of the beta distribution and acquired some of its characteristics as an application in statistics. The outcomes produced here are generic and can give known and novel results.
- extended beta function,
- logarithmic mean,
- extended beta logarithmic distribution
Citation: Mohammed Z. Alqarni, Mohamed Abdalla. Analytic properties and numerical representations for constructing the extended beta function using logarithmic mean[J]. AIMS Mathematics, 2024, 9(5): 12072-12089. doi: 10.3934/math.2024590

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Abstract

This paper aimed to obtain generalizations of both the logarithmic mean ($ \text{L}_{mean} $) and the Euler's beta function (EBF), which we call the extended logarithmic mean ($ \text{EL}_{mean} $) and the extended Euler's beta-logarithmic function (EEBLF), respectively. Also, we discussed various properties, including functional relations, inequalities, infinite sums, finite sums, integral formulas, and partial derivative representations, along with the Mellin transform for the EEBLF. Furthermore, we gave numerical comparisons between these generalizations and the previous studies using MATLAB R2018a in the form of tables and graphs. Additionally, we presented a new version of the beta distribution and acquired some of its characteristics as an application in statistics. The outcomes produced here are generic and can give known and novel results.

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