Probability distribution built by Prabhakar function. Related Turán and Laguerre inequalities

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 Turánian difference for Prabhakar function. Finally, a Laguerre inequality is proved and functional upper bound has been given for Laguerreian difference for Prabhakar function.

KEYWORDS:

• Prabhakar (three-parameter Mittag–Leffler) function
• moments
• Turánian difference
• Laguerre inequality
• functional bounds
• recurrence identities

AMS SUBJECT CLASSIFICATION (2010):

• 26D15
• 33E12
• 60E05
• 62E15

Acknowledgments

Both authors are indebted to the unknown referees and to associate editor S. Yakubovich for several comments and constructive criticisms of the earlier variant of this article, which mainly improve the exposition's relevance and completeness.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Živorad Tomovski acknowledges NWO grant 2015/2016, Department of Applied Mathematics, TU Delft.