INEQUALITIES FOR THE POLYGAMMA FUNCTIONS

Let F-n(x; c) = (Psi((n)) (x))(2) - c Psi((n - 1)) (x)Psi((n+1)) (x) ( x > 0), where Psi denotes the logarithmic derivative of the gamma func tion, n greater than or equal to 2 is an integer, and c is a real numb er. The authors prove that the function x bar right arrow F-n(x; alpha ) is strictly completely monotonic on (0; infinity) if and only if alp ha less than or equal to (n - 1)/n, while x bar right arrow -F-n(x; be ta) is strictly completely monotonic on (0; infinity) if and only if b eta greater than or equal to n/(n + 1).