Monotonicity properties of the gamma function

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