Logarithmically complete monotonicity of ratios of q-gamma functions

For u , v , r , s ∈ R and 0 q ≠ 1 , let Γ q , ψ q be the q-gamma, -psi functions and let W q ; u , v be defined on ( − min ⁡ { u , v } , ∞ ) by W q ; u , v ( x ) = ( Γ q ( x + u ) Γ q ( x + v ) ) 1 / ( u − v ) if u ≠ v and W q ; u , u ( x ) = e ψ q ( x + u ) . In this paper, by a new way we present the necessary and sufficient conditions for the ratio ( W q ; u , v / W q ; r , s ) to be logarithmically completely monotonic on ( − ρ , ∞ ) , where ρ = min ⁡ { u , v , r , s } . This extends and generalizes some existing results.