Some uniqueness results for the non-trivially complete monotonicity of a class of functions involving the polygamma and related functions

Let the function fm, n(x) be given by [image omitted] In the present paper, we use two markedly different methods in order to prove that, among all fm, n(x)(m, n), only f1, 2(x) is non-trivially completely monotonic on (0, ). More precisely, the functions f1, 2(x) and fm, 2n-1(x) are completely monotonic on (0, ), but the functions fm, 2n(x) for (m, n)(1, 1) are not monotonic and do not keep the same sign on (0, ).