Bernoulli and Euler Polynomials - 24.9 Inequalities

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
24.9.E1 | B 2 ⁢ n | > | B 2 ⁢ n ⁡ ( x ) | Bernoulli-number-B 2 𝑛 Bernoulli-polynomial-B 2 𝑛 𝑥 {\displaystyle{\displaystyle|B_{2n}|>|B_{2n}\left(x\right)|}}
|\BernoullinumberB{2n}| > |\BernoullipolyB{2n}@{x}|		 1 > x , x > 0 formulae-sequence 1 𝑥 𝑥 0 {\displaystyle{\displaystyle 1>x,x>0}} 
abs(bernoulli(2*n)) > abs(bernoulli(2*n, x))

Abs[BernoulliB[2*n]] > Abs[BernoulliB[2*n, x]]
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
24.9.E2 ( 2 - 2 1 - 2 ⁢ n ) ⁢ | B 2 ⁢ n | ≥ | B 2 ⁢ n ⁡ ( x ) - B 2 ⁢ n | 2 superscript 2 1 2 𝑛 Bernoulli-number-B 2 𝑛 Bernoulli-polynomial-B 2 𝑛 𝑥 Bernoulli-number-B 2 𝑛 {\displaystyle{\displaystyle(2-2^{1-2n})|B_{2n}|\geq|B_{2n}\left(x\right)-B_{2% n}|}}
(2-2^{1-2n})|\BernoullinumberB{2n}| \geq |\BernoullipolyB{2n}@{x}-\BernoullinumberB{2n}|		 1 ≥ x , x ≥ 0 formulae-sequence 1 𝑥 𝑥 0 {\displaystyle{\displaystyle 1\geq x,x\geq 0}} 
(2 - (2)^(1 - 2*n))*abs(bernoulli(2*n)) >= abs(bernoulli(2*n, x)- bernoulli(2*n))

(2 - (2)^(1 - 2*n))*Abs[BernoulliB[2*n]] >= Abs[BernoulliB[2*n, x]- BernoulliB[2*n]]
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
24.9.E3 4 - n ⁢ | E 2 ⁢ n | > ( - 1 ) n ⁢ E 2 ⁢ n ⁡ ( x ) superscript 4 𝑛 Euler-number-E 2 𝑛 superscript 1 𝑛 Euler-polynomial-E 2 𝑛 𝑥 {\displaystyle{\displaystyle 4^{-n}|E_{2n}|>(-1)^{n}E_{2n}\left(x\right)}}
4^{-n}|\EulernumberE{2n}| > (-1)^{n}\EulerpolyE{2n}@{x}		 

(4)^(- n)*abs(euler(2*n)) > (- 1)^(n)* euler(2*n, x)

(4)^(- n)*Abs[EulerE[2*n]] > (- 1)^(n)* EulerE[2*n, x]
Missing Macro Error Failure Skip - symbolical successful subtest
Failed [4 / 9]
Result: False
Test Values: {Rule[n, 1], Rule[x, 0.5]}


Result: False
Test Values: {Rule[n, 2], Rule[x, 0.5]}

... skip entries to safe data
24.9.E3 ( - 1 ) n ⁢ E 2 ⁢ n ⁡ ( x ) > 0 superscript 1 𝑛 Euler-polynomial-E 2 𝑛 𝑥 0 {\displaystyle{\displaystyle(-1)^{n}E_{2n}\left(x\right)>0}}
(-1)^{n}\EulerpolyE{2n}@{x} > 0		 

(- 1)^(n)* euler(2*n, x) > 0

(- 1)^(n)* EulerE[2*n, x] > 0
Failure Failure
Failed [5 / 9]
Result: 0. < -.7500000000
Test Values: {x = 3/2, n = 1}


Result: 0. < -.1875000000
Test Values: {x = 3/2, n = 2}

... skip entries to safe data
Failed [5 / 9]
Result: False
Test Values: {Rule[n, 1], Rule[x, 1.5]}


Result: False
Test Values: {Rule[n, 2], Rule[x, 1.5]}

... skip entries to safe data
24.9.E4 2 ⁢ ( 2 ⁢ n + 1 ) ! ( 2 ⁢ π ) 2 ⁢ n + 1 > ( - 1 ) n + 1 ⁢ B 2 ⁢ n + 1 ⁡ ( x ) 2 2 𝑛 1 superscript 2 𝜋 2 𝑛 1 superscript 1 𝑛 1 Bernoulli-polynomial-B 2 𝑛 1 𝑥 {\displaystyle{\displaystyle\frac{2(2n+1)!}{(2\pi)^{2n+1}}>(-1)^{n+1}B_{2n+1}% \left(x\right)}}
\frac{2(2n+1)!}{(2\pi)^{2n+1}} > (-1)^{n+1}\BernoullipolyB{2n+1}@{x}		 

(2*factorial(2*n + 1))/((2*Pi)^(2*n + 1)) > (- 1)^(n + 1)* bernoulli(2*n + 1, x)

Divide[2*(2*n + 1)!,(2*Pi)^(2*n + 1)] > (- 1)^(n + 1)* BernoulliB[2*n + 1, x]
Failure Failure Successful [Tested: 3]
Failed [4 / 9]
Result: False
Test Values: {Rule[n, 1], Rule[x, 1.5]}


Result: False
Test Values: {Rule[n, 3], Rule[x, 1.5]}

... skip entries to safe data
24.9.E4 ( - 1 ) n + 1 ⁢ B 2 ⁢ n + 1 ⁡ ( x ) > 0 superscript 1 𝑛 1 Bernoulli-polynomial-B 2 𝑛 1 𝑥 0 {\displaystyle{\displaystyle(-1)^{n+1}B_{2n+1}\left(x\right)>0}}
(-1)^{n+1}\BernoullipolyB{2n+1}@{x} > 0		 

(- 1)^(n + 1)* bernoulli(2*n + 1, x) > 0

(- 1)^(n + 1)* BernoulliB[2*n + 1, x] > 0
Failure Failure
Failed [3 / 3]
Result: 0. < -.3125000000
Test Values: {x = 3/2, n = 2}


Result: 0. < 0.
Test Values: {x = 1/2, n = 2}

... skip entries to safe data
Failed [5 / 9]
Result: False
Test Values: {Rule[n, 2], Rule[x, 1.5]}


Result: False
Test Values: {Rule[n, 1], Rule[x, 0.5]}

... skip entries to safe data
24.9.E5 4 ⁢ ( 2 ⁢ n - 1 ) ! π 2 ⁢ n ⁢ 2 2 ⁢ n - 1 2 2 ⁢ n - 2 > ( - 1 ) n ⁢ E 2 ⁢ n - 1 ⁡ ( x ) 4 2 𝑛 1 superscript 𝜋 2 𝑛 superscript 2 2 𝑛 1 superscript 2 2 𝑛 2 superscript 1 𝑛 Euler-polynomial-E 2 𝑛 1 𝑥 {\displaystyle{\displaystyle\frac{4(2n-1)!}{\pi^{2n}}\frac{2^{2n}-1}{2^{2n}-2}% >(-1)^{n}E_{2n-1}\left(x\right)}}
\frac{4(2n-1)!}{\pi^{2n}}\frac{2^{2n}-1}{2^{2n}-2} > (-1)^{n}\EulerpolyE{2n-1}@{x}		 

(4*factorial(2*n - 1))/((Pi)^(2*n))*((2)^(2*n)- 1)/((2)^(2*n)- 2) > (- 1)^(n)* euler(2*n - 1, x)

Divide[4*(2*n - 1)!,(Pi)^(2*n)]*Divide[(2)^(2*n)- 1,(2)^(2*n)- 2] > (- 1)^(n)* EulerE[2*n - 1, x]
Failure Failure
Failed [1 / 9]
Result: 2.250000000 < .2639824007
Test Values: {x = 2, n = 2}

Failed [1 / 9]
Result: False
Test Values: {Rule[n, 2], Rule[x, 2]}

24.9.E5 ( - 1 ) n ⁢ E 2 ⁢ n - 1 ⁡ ( x ) > 0 superscript 1 𝑛 Euler-polynomial-E 2 𝑛 1 𝑥 0 {\displaystyle{\displaystyle(-1)^{n}E_{2n-1}\left(x\right)>0}}
(-1)^{n}\EulerpolyE{2n-1}@{x} > 0		 

(- 1)^(n)* euler(2*n - 1, x) > 0

(- 1)^(n)* EulerE[2*n - 1, x] > 0
Failure Failure
Failed [7 / 9]
Result: 0. < -1.
Test Values: {x = 3/2, n = 1}


Result: 0. < -.6250000000e-1
Test Values: {x = 3/2, n = 3}

... skip entries to safe data
Failed [7 / 9]
Result: False
Test Values: {Rule[n, 1], Rule[x, 1.5]}


Result: False
Test Values: {Rule[n, 3], Rule[x, 1.5]}

... skip entries to safe data
24.9.E6 5 ⁢ π ⁢ n ⁢ ( n π ⁢ e ) 2 ⁢ n > ( - 1 ) n + 1 ⁢ B 2 ⁢ n 5 𝜋 𝑛 superscript 𝑛 𝜋 𝑒 2 𝑛 superscript 1 𝑛 1 Bernoulli-number-B 2 𝑛 {\displaystyle{\displaystyle 5\sqrt{\pi n}\left(\frac{n}{\pi e}\right)^{2n}>(-% 1)^{n+1}B_{2n}}}
5\sqrt{\pi n}\left(\frac{n}{\pi e}\right)^{2n} > (-1)^{n+1}\BernoullinumberB{2n}		 

5*sqrt(Pi*n)*((n)/(Pi*exp(1)))^(2*n) > (- 1)^(n + 1)* bernoulli(2*n)

5*Sqrt[Pi*n]*(Divide[n,Pi*E])^(2*n) > (- 1)^(n + 1)* BernoulliB[2*n]
Failure Failure
Failed [1 / 3]
Result: .1666666667 < .1215223702
Test Values: {n = 1}

Failed [1 / 3]
Result: False
Test Values: {Rule[n, 1]}

24.9.E6 ( - 1 ) n + 1 ⁢ B 2 ⁢ n > 4 ⁢ π ⁢ n ⁢ ( n π ⁢ e ) 2 ⁢ n superscript 1 𝑛 1 Bernoulli-number-B 2 𝑛 4 𝜋 𝑛 superscript 𝑛 𝜋 𝑒 2 𝑛 {\displaystyle{\displaystyle(-1)^{n+1}B_{2n}>4\sqrt{\pi n}\left(\frac{n}{\pi e% }\right)^{2n}}}
(-1)^{n+1}\BernoullinumberB{2n} > 4\sqrt{\pi n}\left(\frac{n}{\pi e}\right)^{2n}		 

(- 1)^(n + 1)* bernoulli(2*n) > 4*sqrt(Pi*n)*((n)/(Pi*exp(1)))^(2*n)

(- 1)^(n + 1)* BernoulliB[2*n] > 4*Sqrt[Pi*n]*(Divide[n,Pi*E])^(2*n)
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
24.9.E7 8 ⁢ n π ⁢ ( 4 ⁢ n π ⁢ e ) 2 ⁢ n ⁢ ( 1 + 1 12 ⁢ n ) > ( - 1 ) n ⁢ E 2 ⁢ n 8 𝑛 𝜋 superscript 4 𝑛 𝜋 𝑒 2 𝑛 1 1 12 𝑛 superscript 1 𝑛 Euler-number-E 2 𝑛 {\displaystyle{\displaystyle 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}\left(1+\frac{1}{12n}\right) > (-1)^{n}\EulernumberE{2n}		 

8*sqrt((n)/(Pi))*((4*n)/(Pi*exp(1)))^(2*n)*(1 +(1)/(12*n)) > (- 1)^(n)* euler(2*n)

8*Sqrt[Divide[n,Pi]]*(Divide[4*n,Pi*E])^(2*n)*(1 +Divide[1,12*n]) > (- 1)^(n)* EulerE[2*n]
Missing Macro Error Failure - Successful [Tested: 3]
24.9.E7 ( - 1 ) n ⁢ E 2 ⁢ n > 8 ⁢ n π ⁢ ( 4 ⁢ n π ⁢ e ) 2 ⁢ n superscript 1 𝑛 Euler-number-E 2 𝑛 8 𝑛 𝜋 superscript 4 𝑛 𝜋 𝑒 2 𝑛 {\displaystyle{\displaystyle(-1)^{n}E_{2n}>8\sqrt{\frac{n}{\pi}}\left(\frac{4n% }{\pi e}\right)^{2n}}}
(-1)^{n}\EulernumberE{2n} > 8\sqrt{\frac{n}{\pi}}\left(\frac{4n}{\pi e}\right)^{2n}		 

(- 1)^(n)* euler(2*n) > 8*sqrt((n)/(Pi))*((4*n)/(Pi*exp(1)))^(2*n)

(- 1)^(n)* EulerE[2*n] > 8*Sqrt[Divide[n,Pi]]*(Divide[4*n,Pi*E])^(2*n)
Missing Macro Error Failure - Successful [Tested: 3]
24.9.E8 2 ⁢ ( 2 ⁢ n ) ! ( 2 ⁢ π ) 2 ⁢ n ⁢ 1 1 - 2 β - 2 ⁢ n ≥ ( - 1 ) n + 1 ⁢ B 2 ⁢ n ≥ 2 ⁢ ( 2 ⁢ n ) ! ( 2 ⁢ π ) 2 ⁢ n ⁢ 1 1 - 2 - 2 ⁢ n 2 2 𝑛 superscript 2 𝜋 2 𝑛 1 1 superscript 2 𝛽 2 𝑛 superscript 1 𝑛 1 Bernoulli-number-B 2 𝑛 2 2 𝑛 superscript 2 𝜋 2 𝑛 1 1 superscript 2 2 𝑛 {\displaystyle{\displaystyle\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}}}}
\frac{2(2n)!}{(2\pi)^{2n}}\frac{1}{1-2^{\beta-2n}} \geq (-1)^{n+1}\BernoullinumberB{2n}\geq\frac{2(2n)!}{(2\pi)^{2n}}\frac{1}{1-2^{-2n}}		 

(2*factorial(2*n))/((2*Pi)^(2*n))*(1)/(1 - (2)^(beta - 2*n)) >= (- 1)^(n + 1)* bernoulli(2*n) >= (2*factorial(2*n))/((2*Pi)^(2*n))*(1)/(1 - (2)^(- 2*n))

Divide[2*(2*n)!,(2*Pi)^(2*n)]*Divide[1,1 - (2)^(\[Beta]- 2*n)] >= (- 1)^(n + 1)* BernoulliB[2*n] >= Divide[2*(2*n)!,(2*Pi)^(2*n)]*Divide[1,1 - (2)^(- 2*n)]
Failure Failure Error
Failed [2 / 9]
Result: False
Test Values: {Rule[n, 1], Rule[β, 0.5]}


Result: GreaterEqual[DirectedInfinity[], 0.16666666666666666]
Test Values: {Rule[n, 1], Rule[β, 2]}

24.9.E9 β = 2 + ln ⁡ ( 1 - 6 ⁢ π - 2 ) ln ⁡ 2 𝛽 2 1 6 superscript 𝜋 2 2 {\displaystyle{\displaystyle\beta=2+\frac{\ln\left(1-6\pi^{-2}\right)}{\ln 2}}}
\beta = 2+\frac{\ln@{1-6\pi^{-2}}}{\ln@@{2}}		 

beta = 2 +(ln(1 - 6*(Pi)^(- 2)))/(ln(2))

\[Beta] == 2 +Divide[Log[1 - 6*(Pi)^(- 2)],Log[2]]
Aborted Failure
Failed [3 / 3]
Result: .850806174
Test Values: {beta = 3/2}


Result: -.1491938260
Test Values: {beta = 1/2}

... skip entries to safe data
Failed [3 / 3]
Result: 0.850806175200028
Test Values: {Rule[β, 1.5]}


Result: -0.149193824799972
Test Values: {Rule[β, 0.5]}

... skip entries to safe data
24.9.E9 2 + ln ⁡ ( 1 - 6 ⁢ π - 2 ) ln ⁡ 2 = 0.6491 ⁢ … 2 1 6 superscript 𝜋 2 2 0.6491 … {\displaystyle{\displaystyle 2+\frac{\ln\left(1-6\pi^{-2}\right)}{\ln 2}=0.649% 1\dots}}
2+\frac{\ln@{1-6\pi^{-2}}}{\ln@@{2}} = 0.6491\dots		 

2 +(ln(1 - 6*(Pi)^(- 2)))/(ln(2)) = 0.6491

2 +Divide[Log[1 - 6*(Pi)^(- 2)],Log[2]] == 0.6491
Error Failure Skip - symbolical successful subtest Successful [Tested: 1]
24.9.E10 4 n + 1 ⁢ ( 2 ⁢ n ) ! π 2 ⁢ n + 1 > ( - 1 ) n ⁢ E 2 ⁢ n superscript 4 𝑛 1 2 𝑛 superscript 𝜋 2 𝑛 1 superscript 1 𝑛 Euler-number-E 2 𝑛 {\displaystyle{\displaystyle\frac{4^{n+1}(2n)!}{\pi^{2n+1}}>(-1)^{n}E_{2n}}}
\frac{4^{n+1}(2n)!}{\pi^{2n+1}} > (-1)^{n}\EulernumberE{2n}		 

((4)^(n + 1)*factorial(2*n))/((Pi)^(2*n + 1)) > (- 1)^(n)* euler(2*n)

Divide[(4)^(n + 1)*(2*n)!,(Pi)^(2*n + 1)] > (- 1)^(n)* EulerE[2*n]
Missing Macro Error Failure - Successful [Tested: 3]
24.9.E10 ( - 1 ) n ⁢ E 2 ⁢ n > 4 n + 1 ⁢ ( 2 ⁢ n ) ! π 2 ⁢ n + 1 ⁢ 1 1 + 3 - 1 - 2 ⁢ n superscript 1 𝑛 Euler-number-E 2 𝑛 superscript 4 𝑛 1 2 𝑛 superscript 𝜋 2 𝑛 1 1 1 superscript 3 1 2 𝑛 {\displaystyle{\displaystyle(-1)^{n}E_{2n}>\frac{4^{n+1}(2n)!}{\pi^{2n+1}}% \frac{1}{1+3^{-1-2n}}}}
(-1)^{n}\EulernumberE{2n} > \frac{4^{n+1}(2n)!}{\pi^{2n+1}}\frac{1}{1+3^{-1-2n}}		 

(- 1)^(n)* euler(2*n) > ((4)^(n + 1)*factorial(2*n))/((Pi)^(2*n + 1))*(1)/(1 + (3)^(- 1 - 2*n))

(- 1)^(n)* EulerE[2*n] > Divide[(4)^(n + 1)*(2*n)!,(Pi)^(2*n + 1)]*Divide[1,1 + (3)^(- 1 - 2*n)]
Missing Macro Error Failure - Successful [Tested: 3]
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