Probability distribution built by Prabhakar function. Related Tur\'an and Laguerre inequalities

TL;DR

This paper introduces a new probability distribution based on the Prabhakar function, deriving moments and inequalities that extend classical results like Turán and Laguerre inequalities.

Contribution

It establishes explicit moment formulas and bounds for Turán and Laguerre inequalities related to the Prabhakar-based distribution, a novel approach in special function analysis.

Findings

Explicit formulas for raw and factorial moments

Upper bounds for Turán difference of Prabhakar function

Laguerre inequality with functional upper bound

Abstract

Introducing the discrete probability distribution by means of the Prabhakar (or the three--parameter Mittag--Leffler) function, we establish explicit expressions for raw and factorial moments and also general fractional order moments. Applying an elementary moment inequality we obtain functional upper bounds for the Tur\'anian difference for Prabhakar function. Finally, a Laguerre inequality is proved and functional upper bound has been given for Laguerreian difference for Prabhakar function.