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 is given, which is a continuous analogue of a result of Wang and Zhu. On the other hand, the log-behaviour of the functions

is considered, where ζ(x) and Γ(x) are the Riemann zeta function and the Euler Gamma function, respectively. Consequently, the strict log-concavities of the function θ(x) (a conjecture of Chen et al.) and for some combinatorial sequences (including the Bernoulli numbers, the tangent numbers, the Catalan numbers, the Fuss–Catalan numbers, and the binomial coefficients are demonstrated. In particular, this contains some results of Chen et al., and Luca and Stănică. Finally, by researching the logarithmically complete monotonicity of some functions, the infinite log-monotonicity of the sequence

is proved. This generalizes two results of Chen et al. that both the Catalan numbers and the central binomial coefficients are infinitely log-monotonic, and strengthens one result of Su and Wang that is log-convex in n for positive integers d > δ. In addition, the asymptotically infinite log-monotonicity of derangement numbers is showed. In order to research the stronger properties of the above functions θ(x) and F(x), the logarithmically complete monotonicity of functions

is also obtained, which generalizes the results of Lee and Tepedelenlioǧlu, and Qi and Li.

Nous prouvons un encadrement optimal pour la quantité $H\left( x,\,s \right)\,=\,\sum{_{n\ge 1}\,\frac{1}{{{\left( x+n \right)}^{s}}}}$ pour $x\,\ge \,0$ et $s\,>\,1$, qui améliore l'encadrement standard par des intégrales. Cet encadrement entraîne des inégalités sur la fonction $\zeta $ de Riemann, et amène à conjecturer la monotonie de la fonction $s\,\mapsto \,{{[(s\,-\,1)\text{ }\!\!\zeta\!\!\text{ (}s\text{)}]}^{\frac{1}{S-1}}}$. On donne des applications à l'étude de la convexité de fonctions liées à la fonction $\Gamma $ d'Euler et à la majoration optimale des fonctions élémentaires intervenant dans les opérateurs de Baskakov. Puis, nous étendons aux fonctions complètement monotones sur ]$0,\,+\infty $[ les résultats établis pour la fonction $x\,\mapsto \,{{x}^{-s}}$, et nous en déduisons des preuves élémentaires du comportement, quand $z$ tend vers 1, des séries génératrices de certaines fonctions arithmétiques. Enfin, nous prouvons qu'une partie du résultat se généralise à une classe de fonctions convexes positives décroissantes.

It is shown that an elementary pure birth process is a mixed Poisson process iff the sequence of post-jump intensities forms a martingale with respect to the σ -fields generated by the jump times of the process. In this case, the post-jump intensities converge almost surely to the mixing random variable of the process.