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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 (2212n)|B2n| |B2n(x)B2n|,
1x0.

(24.9.3)–(24.9.5) hold for 12>x>0.

24.9.3 4n|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(2n1)!π2n22n122n2 >(1)nE2n1(x)>0.

(24.9.6)–(24.9.7) hold for n=2,3,.

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

Lastly,

24.9.8 2(2n)!(2π)2n112β2n(1)n+1B2n2(2n)!(2π)2n1122n

with

24.9.9 β=2+ln(16π2)ln2=0.6491.
24.9.10 4n+1(2n)!π2n+1>(1)nE2n>4n+1(2n)!π2n+111+312n.