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24 Bernoulli and Euler Polynomials Properties 24.8 Series Expansions 24.10 Arithmetic Properties

§24.9 Inequalities

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Keywords:: Bernoulli numbers, Bernoulli polynomials, Euler numbers, Euler polynomials, inequalities
Notes:: For (24.9.1) and (24.9.2) see Olver (1997b, p. 283). For (24.9.3) see Temme (1996b, p. 16). For (24.9.4) see Lehmer (1940, p. 538). (24.9.5) follows from (24.9.8), with (24.4.35), (24.2.7), and (24.4.26). For (24.9.6) and (24.9.7) see Leeming (1989), and for (24.9.8) see Alzer (2000). (24.9.10) follows from (24.4.28) and (24.8.4) with ; see also Lehmer (1940, p. 538).
Permalink:: http://dlmf.nist.gov/24.9
See also:: Annotations for Ch.24

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

24.9.1 $|B_{2n}|>|B_{2n}\left(x\right)|,$

,

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, : integer and : real or complex
A&S Ref:: 23.1.13
Referenced by:: §24.9
Permalink:: http://dlmf.nist.gov/24.9.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §24.9 and Ch.24

24.9.2 $(2-2^{1-2n})|B_{2n}|\geq|B_{2n}\left(x\right)-B_{2n}|,$ $1\geq x\geq 0$ .

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, : integer and : real or complex
Referenced by:: §24.9
Permalink:: http://dlmf.nist.gov/24.9.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §24.9 and Ch.24

(24.9.3)–(24.9.5) hold for $\tfrac{1}{2}>x>0$ .

24.9.3 $4^{-n}|E_{2n}|>(-1)^{n}E_{2n}\left(x\right)>0,$

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Symbols:: $E_{\NVar{n}}$ : Euler numbers, $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, : integer and : real or complex
A&S Ref:: 23.1.13
Referenced by:: §24.9, §24.9
Permalink:: http://dlmf.nist.gov/24.9.E3
Encodings:: TeX, pMML, png
See also:: Annotations for §24.9 and Ch.24

24.9.4 $\frac{2(2n+1)!}{(2\pi)^{2n+1}}>(-1)^{n+1}B_{2n+1}\left(x\right)>0,$ $n=2,3,\dots$ ,

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Symbols:: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $\pi$ : the ratio of the circumference of a circle to its diameter, : factorial (as in ), : integer and : real or complex
Referenced by:: §24.9
Permalink:: http://dlmf.nist.gov/24.9.E4
Encodings:: TeX, pMML, png
See also:: Annotations for §24.9 and Ch.24

24.9.5 $\frac{4(2n-1)!}{\pi^{2n}}\frac{2^{2n}-1}{2^{2n}-2}>(-1)^{n}E_{2n-1}\left(x% \right)>0.$

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Symbols:: $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $\pi$ : the ratio of the circumference of a circle to its diameter, : factorial (as in ), : integer and : real or complex
A&S Ref:: 23.1.14
Referenced by:: §24.9, §24.9
Permalink:: http://dlmf.nist.gov/24.9.E5
Encodings:: TeX, pMML, png
See also:: Annotations for §24.9 and Ch.24

(24.9.6)–(24.9.7) hold for $n=2,3,\dotsc$ .

24.9.6 $5\sqrt{\pi n}\left(\frac{n}{\pi e}\right)^{2n}>(-1)^{n+1}B_{2n}>4\sqrt{\pi n}% \left(\frac{n}{\pi e}\right)^{2n},$

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm and : integer
Referenced by:: §24.9, §24.9
Permalink:: http://dlmf.nist.gov/24.9.E6
Encodings:: TeX, pMML, png
See also:: Annotations for §24.9 and Ch.24

24.9.7 $8\sqrt{\frac{n}{\pi}}\left(\frac{4n}{\pi e}\right)^{2n}\left(1+\frac{1}{12n}% \right)>(-1)^{n}E_{2n}>8\sqrt{\frac{n}{\pi}}\left(\frac{4n}{\pi e}\right)^{2n}.$

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Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm and : integer
Referenced by:: §24.9, §24.9
Permalink:: http://dlmf.nist.gov/24.9.E7
Encodings:: TeX, pMML, png
See also:: Annotations for §24.9 and Ch.24

Lastly,

24.9.8 $\frac{2(2n)!}{(2\pi)^{2n}}\frac{1}{1-2^{\beta-2n}}\geq(-1)^{n+1}B_{2n}\geq% \frac{2(2n)!}{(2\pi)^{2n}}\frac{1}{1-2^{-2n}}$

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, : factorial (as in ), : integer and $\beta$ : quantity
Referenced by:: §24.9
Permalink:: http://dlmf.nist.gov/24.9.E8
Encodings:: TeX, pMML, png
See also:: Annotations for §24.9 and Ch.24

with

24.9.9 $\beta=2+\frac{\ln\left(1-6\pi^{-2}\right)}{\ln 2}=0.6491\dots.$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\ln\NVar{z}$ : principal branch of logarithm function and $\beta$ : quantity
Permalink:: http://dlmf.nist.gov/24.9.E9
Encodings:: TeX, pMML, png
See also:: Annotations for §24.9 and Ch.24

24.9.10 $\frac{4^{n+1}(2n)!}{\pi^{2n+1}}>(-1)^{n}E_{2n}>\frac{4^{n+1}(2n)!}{\pi^{2n+1}}% \frac{1}{1+3^{-1-2n}}.$

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Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, : factorial (as in ) and : integer
A&S Ref:: 23.1.15
Referenced by:: §24.9
Permalink:: http://dlmf.nist.gov/24.9.E10
Encodings:: TeX, pMML, png
See also:: Annotations for §24.9 and Ch.24

24.8 Series Expansions 24.10 Arithmetic Properties

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