24 Bernoulli and Euler PolynomialsProperties24.8 Series Expansions24.10 Arithmetic Properties

§24.9 Inequalities

ⓘ

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 $x=1/2$; 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 $n=1,2,\mathrm{\dots }$.

24.9.1		$|B_{2n}|$	$>|B_{2n}\left(x\right)|,$
$1>x>0$,
ⓘ Symbols: $B_n$: Bernoulli numbers, $B_n\left(x\right)$: Bernoulli polynomials, $n$: integer and $x$: 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		$(2-2^{1-2n})|B_{2n}|$	$\ge |B_{2n}\left(x\right)-B_{2n}|,$
$1\ge x\ge 0$.
ⓘ Symbols: $B_n$: Bernoulli numbers, $B_n\left(x\right)$: Bernoulli polynomials, $n$: integer and $x$: 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 $\frac{1}{2}>x>0$.

24.9.3		$4^{-n}|E_{2n}|$	$>{(-1)}^n{E}_{2n}\left(x\right)>0,$
ⓘ Symbols: $E_n$: Euler numbers, $E_n\left(x\right)$: Euler polynomials, $n$: integer and $x$: 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		${\displaystyle \frac{2(2n+1)!}{{(2\pi )}^{2n+1}}}$	$>{(-1)}^{n+1}{B}_{2n+1}\left(x\right)>0,$
$n=2,3,\mathrm{\dots }$,
ⓘ Symbols: $B_n\left(x\right)$: Bernoulli polynomials, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $n$: integer and $x$: 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		${\displaystyle \frac{4(2n-1)!}{{\pi }^{2n}}}{\displaystyle \frac{2^{2n}-1}{2^{2n}-2}}$	$>{(-1)}^n{E}_{2n-1}\left(x\right)>0.$
ⓘ Symbols: $E_n\left(x\right)$: Euler polynomials, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $n$: integer and $x$: 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 $n=2,3,\mathrm{\dots }$.

24.9.6		$$5\sqrt{\pi n}{\left(\frac{n}{\pi \mathrm{e}}\right)}^{2n}>{(-1)}^{n+1}{B}_{2n}>4\sqrt{\pi n}{\left(\frac{n}{\pi \mathrm{e}}\right)}^{2n},$$
ⓘ Symbols: $B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm and $n$: 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		$$8\sqrt{\frac{n}{\pi }}{\left(\frac{4n}{\pi \mathrm{e}}\right)}^{2n}\left(1+\frac{1}{12n}\right)>{(-1)}^n{E}_{2n}>8\sqrt{\frac{n}{\pi }}{\left(\frac{4n}{\pi \mathrm{e}}\right)}^{2n}.$$
ⓘ Symbols: $E_n$: Euler numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm and $n$: 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		$$\frac{2(2n)!}{{(2\pi )}^{2n}}\frac{1}{1-2^{\beta -2n}}\ge {(-1)}^{n+1}{B}_{2n}\ge \frac{2(2n)!}{{(2\pi )}^{2n}}\frac{1}{1-2^{-2n}}$$
ⓘ Symbols: $B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $n$: 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		$$\beta =2+\frac{\ln \left(1-6{\pi }^{-2}\right)}{\ln 2}=0.6491\mathrm{\dots }.$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: 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		$$\frac{4^{n+1}(2n)!}{{\pi }^{2n+1}}>{(-1)}^n{E}_{2n}>\frac{4^{n+1}(2n)!}{{\pi }^{2n+1}}\frac{1}{1+3^{-1-2n}}.$$
ⓘ Symbols: $E_n$: Euler numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$) and $n$: 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 Expansions24.10 Arithmetic Properties

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