Some classes of completely monotonic functions

We prove: (i) Let F-n(x) = P-n(x) [e- (1+ 1/x)(x)] and G(n)(x) =P-n(x) (1 + 1/x)(x+1) - e], where P-n(x) = x(n) + Sigma(nu=0)(n-1) c(nu)x(nu) is a polynomial of degree n greater than or equal to 1 with real coefficients. F-n is completely monotonic if and only if n = 1 and co greater than or equal to 11/12; and G(n) is completely monotonic if and only if n = 1 and c(o) greater than or equal to 1/12. (ii) The functions x bar right arrow e - (1 + 1/x)(x) and x bar right arrow (1 + 1/x)(x+1) - e are Stieltjes transforms and in particular they are completely monotonic. (iii) Let a > 0 and b be real numbers. The function x bar right arrow (1 + a/x)(x+b) - e(a) is completely monotonic if and only if a less than or equal to 2b. Part (i) extends and complements a recently published result of Sandor and Debnath, while part (iii) generalizes a theorem of Schur.