Logarithmically complete monotonicity of ratios of q-gamma functions

Abstract

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