Inequalities for the gamma function

@inproceedings{Alzer2000InequalitiesFT,
  title={Inequalities for the gamma function},
  author={H. Alzer},
  year={2000}
}
  • H. Alzer
  • Published 2000
  • Mathematics
  • We prove the following two theorems: (i) Let Mr(a, b) be the rth power mean of a and b. The inequality Mr(Γ(x), Γ(1/x)) ≥ 1 holds for all x ∈ (0,∞) if and only if r ≥ 1/C − π2/(6C2), where C denotes Euler’s constant. This refines results established by W. Gautschi (1974) and the author (1997). (ii) The inequalities xα(x−1)−C < Γ(x) < xβ(x−1)−C (∗) are valid for all x ∈ (0, 1) if and only if α ≤ 1−C and β ≥ (π2/6−C)/2, while (∗) holds for all x ∈ (1,∞) if and only if α ≤ (π2/6− C)/2 and β ≥ 1… CONTINUE READING
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