Some inequalities involving two generalized beta functions in n variables

Mustapha Raïssouli,Salma I El-Soubhy

doi:10.1186/s13660-021-02625-8

Abstract

The beta and gamma functions have recently seen several developments and various extensions because of their nice properties and interesting applications. The contribution of this paper falls within this framework. After introducing a generalized gamma function and two generalized beta functions in several variables, we investigate some inequalities involving these generalized functions.

Highlights

The standard beta and gamma functions are respectively defined by∀x, y > 0, ∀x > 0, B(x, y) =: tx–1(1 – t)y–1 dt, +∞(x) =: tx–1e–t dt.Such functions play an important role in mathematical analysis and have wide applications in various contexts of mathematics and physics
The extension of the beta function from two to n variables has been introduced in the literature [1, 2, 6, 21, 22]
For a systematic study of the properties of B(x1, . . . , xn; a), as well as those of B(x1, . . . , xn; a1, . . . , an), one can consult [22]. This manuscript will be organized as follows: In Sect. 2 we introduce the generalized gamma function in n variables in a brief and simple setting

Summary

The standard beta and gamma functions are respectively defined by

In [7], Chaudhry et al introduced the following extended beta function:. In [19], Özergin et al introduced the generalized beta and gamma functions defined respectively by. The extension of the beta function from two to n variables has been introduced in the literature [1, 2, 6, 21, 22]. 2 we introduce the generalized gamma function in n variables in a brief and simple setting.

If p

Then we get

It is then enough to find an upper bound of t