THE MONOTONICITY AND LOG-BEHAVIOUR OF SOME FUNCTIONS RELATED TO THE EULER GAMMA FUNCTION

The aim of this paper is to develop analytic techniques to deal with the monotonicity of certain combinatorial sequences. On the one hand, a criterion for the monotonicity of the function (x)root f(x) is given, which is a continuous analogue of a result of Wang and Zhu. On the other hand, the log-behaviour of the functions theta(x) = x root 2 zeta(x)Gamma(x + 1) and F(x) = x root Gamma(ax + b + 1)/Gamma(cx + d + 1)Gamma(ex + f + 1) is considered, where zeta( x) and Gamma(x) are the Riemann zeta function and the Euler Gamma function, respectively. Consequently, the strict log-concavities of the function theta(x) (a conjecture of Chen et al.) and {(n)root z(n)} for some combinatorial sequences (including the Bernoulli numbers, the tangent numbers, the Catalan numbers, the Fuss-Catalan numbers, and the binomial coefficients ((2n)(n) ), ((3n)(n) ), ((4n)(n) ), ((5n)(n) ), ((5n)(2n) )) are demonstrated. In particular, this contains some results of Chen et al., and Luca and Stanica. Finally, by researching the logarithmically complete monotonicity of some functions, the infinite log-monotonicity of the sequence { (n(0) + ia)!/(k(0) + ib)! ((k) over bar (0) + i (b) over bar)!}(i >= 0) is proved. This generalizes two results of Chen et al. that both the Catalan numbers (1/(n+ 1)) ((2n)(n)) and the central binomial coefficients ((2n)(n)) are infinitely log- monotonic, and strengthens one result of Su and Wang that ((delta n)(dn)) is log- convex in n for positive integers d > delta. In addition, the asymptotically infinite log- monotonicity of derangement numbers is showed. In order to research the stronger properties of the above functions 0(x) and F(x), the logarithmically complete monotonicity of functions 1/(x)root a zeta(x + b)Gamma(x + c) and x root rho(n)Pi(i=1) Gamma(x + a(i))/Gamma(x + b(i)) is also obtained, which generalizes the results of Lee and Tepedelenlioglu, and Qi and Li.