§24.9 Inequalities

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