Numerical and graphical exploration of the generalized beta-logarithmic matrix function and its properties

This paper investigates the generalized beta-logarithmic matrix function (GBLMF),which combines the extended beta matrix function and the logarithmic mean. The study establishes essential properties of this function, including functional relations, inequalities, finite and infinite sums, integral representations, and partial derivative formulas. Theoretical results are accompanied by numerical examples and graphical representations to demonstrate the behavior of the new matrix function. Additionally, a comparison with classical and previously studied beta matrix functions is presented to highlight the differences and advantages of the generalized version. The findings offer valuable insights into the properties and applications of the extended beta-logarithmic matrix function in various mathematical and applied contexts.