On Gautschi's harmonic mean inequality for the gamma function

  • Authors:

  • Horst Alzer

  • Affiliations:

  • Morsbacher Strasse 10, D-51545 Waldbröl, Germany

  • Venue:
  • Journal of Computational and Applied Mathematics
  • Year:
  • 2003

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Abstract

Let Hn = infx∈Sn (1/n Σk=1n 1/Γ(xk))-1, where Sn = {(x1,..., xn) ∈ R+n : Πk=1n xk = 1}. Gautschi (SIAM J. Math. Anal. 5 (1974)) showed that H2 = 1 and Hn n ≥ 9. In this paper we prove his conjecture that Hn = 1 for n ≤ 8.