On Extended Beta Function and Related Inequalities

Rakesh K. Parmar, Tibor K. Pogány () and Ljiljana Teofanov
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Rakesh K. Parmar: Department of Mathematics, Ramanujan School of Mathematical Sciences, Pondicherry University, Puducherry 605014, India
Tibor K. Pogány: Institute of Applied Mathematics, John von Neumann Faculty of Informatics, Óbuda University, 1034 Budapest, Hungary
Ljiljana Teofanov: Faculty of Technical Sciences, University of Novi Sad, 21000 Novi Sad, Serbia

Mathematics, 2024, vol. 12, issue 17, 1-10

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 K ν + 1 2 p t − 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 K ν + 1 2 .

Keywords: extended Beta function; incomplete extended Beta function; functional upper and lower bounds; Macdonald function; iteration method; extended Beta probability distribution (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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