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

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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.

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