Logarithmically complete monotonicity of ratios of q-gamma functions

For u, v, r, s is an element of R and 0 < q not equal 1, let Gamma(q) , psi(q) be the q-gamma,-psi functions and let W-q;u,W-v be defined on (- min {u, v} , infinity) by W-q;u,W-v (x) =(Gamma(q) (x + u)/Gamma(q)(x + v))(1/(u - v)) = if u not equal v and W-q;u,W-u (x) = e(psi q(x+u).) In this paper, by a new way we present the necessary and sufficient conditions for the ratio (W-q;u,W-v/W-q;r,W-s) to be logarithmically completely monotonic on (-rho, infinity), where rho = min {u, v, r, s}. This extends and generalizes some existing results. (c) 2021 Elsevier Inc. All rights reserved.