Some inequalities involving two generalized beta functions in n variables

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.

The Beta‐Pochhammer and its application to arbitrary threshold phase error probability of a vector in Gaussian noise

“…The Beta‐Pochhammer symbol could be extended to any number of variables. For instance, we can develop the three‐variable β‐Pochhammer using (see, e.g., Rehman et al [9] and Raissouli and El‐Soubhy [10]) so that These ( and ) may be combined to give This could be used in Lauricella functions of three or more variables (e.g., Slater [56] (p. 227)). One of the simplest Lauricella functions of zeroth form employing () is given by the following.…”

“…The Beta-Pochhammer symbol could be extended to any number of variables. For instance, we can develop the three-variable β-Pochhammer using (see, e.g., Rehman et al [9] and Raissouli and El-Soubhy [10])…”

“…Extensions of the Pochhammer symbol have previously been proposed including the q ‐generalization [5], the p ‐extension [6], and additional generalisations of both the Gamma and Beta functions [7]. The k ‐Pochhammer symbol was introduced by Diaz and Pariguan [8] based on the Gamma weighting 𝑡 𝑎−1 𝑒 −𝑡 with further generalizations by, for example, Rehman et al [9], Raissouli and El‐Soubhy [10], and Saboor et al [11]. Further, a wide range of extended Pochhammer symbols based on the Beta weighting 𝑡 𝑎−1 (1 − 𝑡) 𝑏−1 have been presented by authors such as Marfaing [12], Chand et al [13], Srivastava et al [14], Abubakar et al [15], Palsaniya et al [16], and Ghanim and Al‐Janaby [17].…”