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

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.