Monotonicity properties of the gamma function

Let Gc(x)=logΓ(x)−xlogx+x−12log(2π)+12ψ(x+c)(x>0;c≥0). We prove that GaGa is completely monotonic on (0,∞)(0,∞) if and only if a≥1/3a≥1/3. Also, −Gb−Gb is completely monotonic on (0,∞)(0,∞) if and only if b=0b=0. An application of this result reveals that the best possible nonnegative constants α,βα,β in 2πxxexp(−x−12ψ(x+α))<Γ(x)<2πxxexp(−x−12ψ(x+β))(x>0) are given by α=1/3α=1/3 and β=0β=0.