Necessary and sufficient conditions for a class of functions and their reciprocals to be logarithmically completely monotonic

We prove that the function F-alpha,F-beta(x) = x(alpha)G(beta)(x)/Gamma(beta x) is strictly logarithmically completely monotonic on (0, infinity) if and only if (alpha, beta) is an element of {(alpha, beta) : beta > 0, beta >= 2 alpha + 1, beta >= alpha + 1}\{(alpha, beta) : alpha = 0, beta = 1} and that [F-alpha,F-beta(x)](-1) is strictly logarithmically completely monotonic on (0, infinity) if and only if (alpha, beta) is an element of {(alpha, beta) : beta > 0, beta <= 2 alpha + 1, beta <= alpha + 1} \{(alpha, beta) : alpha = 0, beta = 1}.