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Keywords:

  • Turán inequality;
  • Bernoulli numbers;
  • Euler numbers;
  • generating functions;
  • remainders in asymptotic expansions;
  • completely monotonic functions of positive order;
  • ratio of series of functions;
  • β-function;
  • MSC (2010) Primary: 11B68;
  • Secondary: 11B83;
  • 33E20;
  • 26D07;
  • 26D15;
  • 41A60;
  • 41A80

Abstract

Turán type inequalities for the partial sums of the generating functions of the Bernoulli and Euler numbers are established. They are shown to follow from a general result relating Turán inequalities of partial sums with Turán inequalities of the corresponding remainders in any Maclaurin expansion. Remainders in asymptotic expansions of the β-function are shown to be completely monotonic of positive order.