Convesity, complete monotonicity and inequalities on zeta and gamma functions, on functions of Baskov operators and on arithmetic functions

We give optimal upper and lower bounds for the function H(x, s) = Sigma(ngreater than or equal to1) 1/(x+n)(s), for x greater than or equal to 0 and s > 1. These bounds improve the standard inequalities with integrals. We deduce from them inequalities about Riemann's zeta function, and we give a conjecture about the monotonicity of the function s --> [(s-1)zeta(s)](1/s-1) Some applications concern the convexity of functions related to Euler's F function and optimal majorization of elementary functions of Baskakov's operators. Then, the result proved for the function x --> x(-s) is extended to completely monotonic functions. This leads to easy evaluation of the order of the generating series of some arithmetical functions when z tends to 1. The last part is concerned with the class of non negative decreasing convex functions on ]0, +infinity[, integrable at infinity.