Inequalities for the Tails of Some Elementary Series

We prove that if Dn, k(x) = Γ(n − k + 2) Γ(n + k + 2) In−k(x) In+k(x) − [Γ(n + 2) In(x)]2 with [formula] then we have for all x > 0, Dn, k(x) < Dn, k+1(x) (n ≥ k ≥ 0),Dn, k+1(x) < 12[Dn, k(x) + Dn, k+2(x)] (n > k ≥ 0), and [Dn+1, k(x)]2 < Dn, k(x) Dn+2, k(x) (n ≥ k − 1 ≥ 0)