SOME LOGARITHMICALLY COMPLETELY MONOTONIC FUNCTIONS AND INEQUALITIES FOR MULTINOMIAL COEFFICIENTS AND MULTIVARIATE BETA FUNCTIONS

Feng Qi (1, 2, 3, 4) , Da-Wei Niu (5) , Dongkyu Lim (6) , Bai-Ni Guo (1)

1 Henan Polytechnic University
2 Tianjin Polytechnic University
3 Inner Mongolia University for Nationalities
4 UESTC - University of Electronic Science and Technology of China [Chengdu]
5 ECNU - East China Normal University [Shangaï]
6 SKKU - Sungkyunkwan University [Suwon]

In the paper, the authors extend a function arising from the Bernoulli trials in probability and involving the gamma function to its largest ranges, find logarithmically complete monotonicity of these extended functions, and, in the light of logarithmically complete monotonicity of these extended functions, derive some inequalities for multinomial coefficients and multivari-ate beta functions. These results recover, extend, and generalize some known conclusions.

Mots clés: logarithmically complete monotonicity; extension; gamma function; combinatorial inequality; multinomial coefficient; multivariate beta function; Bernoulli trial; binomial probability

Domaines: Combinatoire [math.CO]; Théorie des nombres [math.NT]; Analyse classique [math.CA]

Feng Qi, Da-Wei Niu, Dongkyu Lim, Bai-Ni Guo. SOME LOGARITHMICALLY COMPLETELY MONOTONIC FUNCTIONS AND INEQUALITIES FOR MULTINOMIAL COEFFICIENTS AND MULTIVARIATE BETA FUNCTIONS: Completely monotonic functions and inequalities. 2018. ⟨hal-01769288v1⟩