Inequalities for the constants of Landau and Lebesgue

  • Authors:

  • Horst Alzer

  • Affiliations:

  • Morsbacher Str. 10, D-51545 Waldbröl, Germany

  • Venue:
  • Journal of Computational and Applied Mathematics
  • Year:
  • 2002

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Abstract

The constants of Landau and Lebesgue are defined for all integers n ≥ 0 by Gn = Σk=0n1/16k )2 and Ln = 12/π ∫-ππ| sin((n+½)t)/sin(½t)| dt, respectively. We establish sharp inequalities for Gn and Ln/2 in terms of the logarithmic derivative of the gamma function. Further, we prove that the sequence (ΔGn) is completely monotonic, we provide best possible upper and lower bounds for the ratios (Gn-1 + Gn+1)/Gn and (L(n-1)/2+L(n+1)/2)/Ln/2, and we present sharp bounds for Ln/2/Gn and Ln/2 - Gn.