On Extended Beta Function and Related Inequalities

Abstract

In this article, we consider a unified generalized version of extended Euler’s Beta function’s integral form a involving Macdonald function in the kernel. Moreover, we establish functional upper and lower bounds for this extended Beta function. Here, we consider the most general case of the four-parameter Macdonald function Kν+12pt−λ+q(1−t)−μ when λ≠μ in the argument of the kernel. We prove related bounding inequalities, simultaneously complementing the recent results by Parmar and Pogány in which the extended Beta function case λ=μ is resolved. The main mathematical tools are integral representations and fixed-point iterations that are used for obtaining the stationary points of the argument of the Macdonald kernel function Kν+12.

Keywords

• extended Beta function
• incomplete extended Beta function
• functional upper and lower bounds
• Macdonald function
• iteration method
• extended Beta probability distribution