Inequalities And Complete Monotonicity Properties For The Gamma Function And The Barnes G-function

In this paper, we prove some (logarithmically) completely monotonic functions involvingthe gamma function, and establish new upper and lower bounds for Barnes G-function. Themain results as follows:1. It was proved that the functionis strictly decreasing and convex on (0,∞), see Theorem 1.16 of the paper by Qiu Songliangand Vuorinen [Some properties of the gamma and psi functions with applications, Math.Comp., 2004, 74(250): 723-742]. It was showed that this function is logarithmically completelymonotonic on (0,∞), see Theorem 3 of the paper by Chen Chaoping [Complete monotonicityand logarithmically complete monotonicity properties for the gamma and psi functions, J.Math. Anal. Appl., 2007, 336(2): 812-822]. We consider a more general result: Forα,β> 0,We consider the logarithmically complete monotonicity of the function fα,βon (0,∞).2. (i) The following open problem was proposed by Qi Feng and Guo Baini [Completemonotonicities of functions involving the gamma and the digamma functions, RGMIA Res.Rep. Col., 2004, 7(1), Art. 8]: Find conditions aboutαandβsuch that the function is completely (absolutely) monotonic on (?1,∞). We consider the logarithmically completemonotonicity of this function.(ii) We consider a more general result on (0,∞): Forα≥0,β≥0,c≥0, letWe consider the logarithmically complete monotonicity of the function G on (0,∞).3. (i) Forα> 0,β∈R, let(ii) Forα≥0,τ≥0,β∈R, let(iii) Forα> 0,c∈R,β∈R, letWe consider the logarithmically complete monotonicity of these three functions.4. We establish new upper and lower bounds for Barnes G-function.