Inequalities For Gamma And Q -Gamma Functions Of Complex Arguments

Keywords

bounds; Complete monotonicity; gamma function; infinite divisibility; Lerch's inequality; q -gamma function

Abstract

We prove that the function Γ (x + a)Γ (x + b)/|Γ (x + c + iy)|2,a + b = 2c, and its q-analogue are of the form e-h(x,y) and h is completely monotonic in x. In particular both Γ (x + a)Γ (x + b)/|Γ (x + c + iy)|2 and Γ q(x + a)Γ q(x + b)/|Γ q(x + c + iy)|2 are Laplace transforms of infinitely divisible distributions. We also extend Lerch's inequality to the q-gamma function.

Publication Date

9-1-2017

Publication Title

Analysis and Applications

Volume

15

Issue

5

Number of Pages

641-651

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1142/S0219530516500093

Socpus ID

84973151643 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/84973151643

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