# §24.9 Inequalities

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

 24.9.1 $\displaystyle|B_{2n}|$ $\displaystyle>|B_{2n}\left(x\right)|,$ $1>x>0$, ⓘ Symbols: $B_{\NVar{n}}$: Bernoulli numbers, $B_{\NVar{n}}\left(\NVar{x}\right)$: 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 $\displaystyle(2-2^{1-2n})|B_{2n}|$ $\displaystyle\geq|B_{2n}\left(x\right)-B_{2n}|,$ $1\geq x\geq 0$. ⓘ Symbols: $B_{\NVar{n}}$: Bernoulli numbers, $B_{\NVar{n}}\left(\NVar{x}\right)$: 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 $\tfrac{1}{2}>x>0$.

 24.9.3 $\displaystyle 4^{-n}|E_{2n}|$ $\displaystyle>(-1)^{n}E_{2n}\left(x\right)>0,$ ⓘ Symbols: $E_{\NVar{n}}$: Euler numbers, $E_{\NVar{n}}\left(\NVar{x}\right)$: 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 $\displaystyle\frac{2(2n+1)!}{(2\pi)^{2n+1}}$ $\displaystyle>(-1)^{n+1}B_{2n+1}\left(x\right)>0,$ $n=2,3,\dots$, 24.9.5 $\displaystyle\frac{4(2n-1)!}{\pi^{2n}}\frac{2^{2n}-1}{2^{2n}-2}$ $\displaystyle>(-1)^{n}E_{2n-1}\left(x\right)>0.$ ⓘ 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 $n!$), $n$: 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 $n=2,3,\dotsc$.

 24.9.6 $5\sqrt{\pi n}\left(\frac{n}{\pi e}\right)^{2n}>(-1)^{n+1}B_{2n}>4\sqrt{\pi n}% \left(\frac{n}{\pi e}\right)^{2n},$ ⓘ 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 $n$: 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 $8\sqrt{\frac{n}{\pi}}\left(\frac{4n}{\pi e}\right)^{2n}\left(1+\frac{1}{12n}% \right)>(-1)^{n}E_{2n}>8\sqrt{\frac{n}{\pi}}\left(\frac{4n}{\pi e}\right)^{2n}.$ ⓘ 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 $n$: 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 $\frac{2(2n)!}{(2\pi)^{2n}}\frac{1}{1-2^{\beta-2n}}\geq(-1)^{n+1}B_{2n}\geq% \frac{2(2n)!}{(2\pi)^{2n}}\frac{1}{1-2^{-2n}}$

with

 24.9.9 $\beta=2+\frac{\ln\left(1-6\pi^{-2}\right)}{\ln 2}=0.6491\dots.$ ⓘ 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 $\frac{4^{n+1}(2n)!}{\pi^{2n+1}}>(-1)^{n}E_{2n}>\frac{4^{n+1}(2n)!}{\pi^{2n+1}}% \frac{1}{1+3^{-1-2n}}.$ ⓘ Symbols: $E_{\NVar{n}}$: 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