Digital Library of Mathematical Functions
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24 Bernoulli and Euler PolynomialsProperties

§24.9 Inequalities

Except where otherwise noted, the inequalities in this section hold for n=1,2,.

24.9.1 |B2n| >|B2n(x)|,
1>x>0,
24.9.2 (2-21-2n)|B2n| |B2n(x)-B2n|,
1x0.

(24.9.3)–(24.9.5) hold for 12>x>0.

24.9.3 4-n|E2n| >(-1)nE2n(x)
>0,
24.9.4 2(2n+1)!(2π)2n+1 >(-1)n+1B2n+1(x)
>0,
n=2,3,,
24.9.5 4(2n-1)!π2n22n-122n-2 >(-1)nE2n-1(x)
>0.

(24.9.6)–(24.9.7) hold for n=2,3,.

24.9.6 5πn(nπ)2n>(-1)n+1B2n>4πn(nπ)2n,
24.9.7 8nπ(4nπ)2n(1+112n)>(-1)nE2n>8nπ(4nπ)2n.

Lastly,

24.9.8 2(2n)!(2π)2n11-2β-2n(-1)n+1B2n2(2n)!(2π)2n11-2-2n

with

24.9.9 β=2+ln(1-6π-2)ln2=0.6491.
24.9.10 4n+1(2n)!π2n+1>(-1)nE2n>4n+1(2n)!π2n+111+3-1-2n.