Logarithmically Complete Monotonicity Properties Relating to the Gamma Function

We prove that the function f α , β ( x ) = Γ β ( x + α )/ x α Γ( β x ) is strictly logarithmically completely monotonic on (0, ∞) if and [ f α , β ( x )] −1 is strictly logarithmically completely monotonic on (0, ∞) if , where φ 1 ( α , β ) = ( α 2 + α − 1) β 2 + (2 α 2 − 3 α + 1) β − α and φ 2 ( α , β ) = ( α − 1) β 2 + (2 α 2 − 5 α + 2) β − 1.