A class of completely monotonic functions involving the polygamma functions

Let Gamma(x) denote the classical Euler gamma function. We set psi(n)(x) = (-1)(n-1) psi((n))(x) (n is an element of N), where psi((n))(x) denotes the nth derivative of the psi function psi((x)) = Gamma'(x)/Gamma(x). For lambda, alpha, beta is an element of R and m,n is an element of N, we establish necessary and sufficient conditions for the functions L(x;lambda, alpha , beta)= psi(m+n)(x) - lambda psi(m)(x + alpha)psi(n)(x + beta) and -L(x;lambda, alpha, beta) to be completely monotonic on (- min(alpha, beta, 0),infinity). As a result, we generalize and refine some inequalities involving the polygamma functions and also give some inequalities in terms of the ratio of gamma functions.

Automated summary

In this article, we investigate the monotonicity of a class of functions and provide inequalities involving polygamma functions and the ratio of gamma functions.