§24.9 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).
Keywords:: Bernoulli numbers, Bernoulli polynomials, Euler numbers, Euler polynomials, inequalities
Permalink:: http://dlmf.nist.gov/24.9

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

24.9.1 $|\mathop{B_{{2n}}\/}\nolimits|>|\mathop{B_{{2n}}\/}\nolimits\!\left(x\right)|,$ $1>x>0$ ,

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : 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

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

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : 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

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

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

Symbols:: $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : 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

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

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $!$ : $n!$ : factorial, $n$ : integer and $x$ : real or complex
Referenced by:: §24.9
Permalink:: http://dlmf.nist.gov/24.9.E4
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, $!$ : $n!$ : factorial, $n$ : 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

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

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

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $e$ : base of exponential function and $n$ : integer
Referenced by:: §24.9, §24.9
Permalink:: http://dlmf.nist.gov/24.9.E6
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, $e$ : base of exponential function and $n$ : integer
Referenced by:: §24.9, §24.9
Permalink:: http://dlmf.nist.gov/24.9.E7
Encodings:: TeX, pMML, png

Lastly,

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

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $!$ : $n!$ : factorial, $n$ : integer and $\beta$ : quantity
Referenced by:: §24.9
Permalink:: http://dlmf.nist.gov/24.9.E8
Encodings:: TeX, pMML, png

with

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

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $\beta$ : quantity
Permalink:: http://dlmf.nist.gov/24.9.E9
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, $!$ : $n!$ : factorial 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