Monotonicity properties of the gamma function

@article{Alzer2007MonotonicityPO,
  title={Monotonicity properties of the gamma function},
  author={H. Alzer and Necdet Batir},
  journal={Appl. Math. Lett.},
  year={2007},
  volume={20},
  pages={778-781}
}
  • H. Alzer, Necdet Batir
  • Published 2007
  • Mathematics, Computer Science
  • Appl. Math. Lett.
  • 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 . 
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