Logarithmically Complete Monotonicity Properties Relating to the Gamma Function

Figure 1

The blue curve is the graph of the equation 𝜑₁(𝑥,𝑦)=0 with the vertical asymptotic line 𝑥=(√5−1)/2 and the green curve is the graph of 𝜑₂(𝑥,𝑦)=0 with the vertical asymptotic line 𝑥=1.