Turán type inequalities for the partial sums of the generating functions of Bernoulli and Euler numbers

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.