Some logarithmically completely monotonic functions and inequalities for multinomial coefficients and multivariate beta functions

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

1 Tianjin Polytechnic University
2 Inner Mongolia University for Nationalities
3 Henan University of Animal Husbandry and Economy
4 Andong National University
5 Henan Polytechnic University

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 light of logarithmically complete monotonicity of these extended functions, derive some inequalities for multinomial coefficients and multivariate beta functions. These results recover, extend, and generalize some known conclusions.

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

Domaines: Combinatoire [math.CO]; Théorie des nombres [math.NT]; Analyse classique [math.CA]; Théorie [stat.TH]; Probabilités [math.PR]; Statistiques [math.ST]

Feng Qi, Da-Wei Niu, Dongkyu Lim, Bai-Ni Guo. Some logarithmically completely monotonic functions and inequalities for multinomial coefficients and multivariate beta functions. Applicable Analysis and Discrete Mathematics, 2020, 14 (2), pp.512--527. ⟨10.2298/AADM191111033Q⟩. ⟨hal-01769288v3⟩