Elementary Functions - 4.5 Inequalities

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
4.5.E1 x 1 + x < ln ⁑ ( 1 + x ) π‘₯ 1 π‘₯ 1 π‘₯ {\displaystyle{\displaystyle\frac{x}{1+x}<\ln\left(1+x\right)}}
\frac{x}{1+x} < \ln@{1+x}		 x > - 1 , x β‰  0 formulae-sequence π‘₯ 1 π‘₯ 0 {\displaystyle{\displaystyle x>-1,x\neq 0}} 
(x)/(1 + x) < ln(1 + x)

Divide[x,1 + x] < Log[1 + x]
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
4.5.E1 ln ⁑ ( 1 + x ) < x 1 π‘₯ π‘₯ {\displaystyle{\displaystyle\ln\left(1+x\right)<x}}
\ln@{1+x} < x		 x > - 1 , x β‰  0 formulae-sequence π‘₯ 1 π‘₯ 0 {\displaystyle{\displaystyle x>-1,x\neq 0}} 
ln(1 + x) < x

Log[1 + x] < x
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
4.5.E2 x < - ln ⁑ ( 1 - x ) π‘₯ 1 π‘₯ {\displaystyle{\displaystyle x<-\ln\left(1-x\right)}}
x < -\ln@{1-x}		 x < 1 , x β‰  0 formulae-sequence π‘₯ 1 π‘₯ 0 {\displaystyle{\displaystyle x<1,x\neq 0}} 
x < - ln(1 - x)

x < - Log[1 - x]
Failure Failure Successful [Tested: 1] Successful [Tested: 1]
4.5.E2 - ln ⁑ ( 1 - x ) < x 1 - x 1 π‘₯ π‘₯ 1 π‘₯ {\displaystyle{\displaystyle-\ln\left(1-x\right)<\frac{x}{1-x}}}
-\ln@{1-x} < \frac{x}{1-x}		 x < 1 , x β‰  0 formulae-sequence π‘₯ 1 π‘₯ 0 {\displaystyle{\displaystyle x<1,x\neq 0}} 
- ln(1 - x) < (x)/(1 - x)

- Log[1 - x] < Divide[x,1 - x]
Failure Failure Successful [Tested: 1] Successful [Tested: 1]
4.5.E3 | ln ⁑ ( 1 - x ) | < 3 2 ⁒ x 1 π‘₯ 3 2 π‘₯ {\displaystyle{\displaystyle|\ln\left(1-x\right)|<\tfrac{3}{2}x}}
|\ln@{1-x}| < \tfrac{3}{2}x		 0 < x , x ≀ 0.5828 ⁒ … formulae-sequence 0 π‘₯ π‘₯ 0.5828 … {\displaystyle{\displaystyle 0<x,x\leq 0.5828\dots}} 
abs(ln(1 - x)) < (3)/(2)*x

Abs[Log[1 - x]] < Divide[3,2]*x
Failure Failure Successful [Tested: 1] Successful [Tested: 1]
4.5.E4 ln ⁑ x ≀ x - 1 π‘₯ π‘₯ 1 {\displaystyle{\displaystyle\ln x\leq x-1}}
\ln@@{x} \leq x-1		 x > 0 π‘₯ 0 {\displaystyle{\displaystyle x>0}} 
ln(x) <= x - 1

Log[x] <= x - 1
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
4.5.E5 ln ⁑ x ≀ a ⁒ ( x 1 / a - 1 ) π‘₯ π‘Ž superscript π‘₯ 1 π‘Ž 1 {\displaystyle{\displaystyle\ln x\leq a(x^{1/a}-1)}}
\ln@@{x} \leq a(x^{1/a}-1)		 a > 0 , x > 0 formulae-sequence π‘Ž 0 π‘₯ 0 {\displaystyle{\displaystyle a>0,x>0}} 
ln(x) <= a*((x)^(1/a)- 1)

Log[x] <= a*((x)^(1/a)- 1)
Error Failure - Successful [Tested: 9]
4.5.E6 | ln ⁑ ( 1 + z ) | ≀ - ln ⁑ ( 1 - | z | ) 1 𝑧 1 𝑧 {\displaystyle{\displaystyle|\ln\left(1+z\right)|\leq-\ln\left(1-|z|\right)}}
|\ln@{1+z}| \leq -\ln@{1-|z|}		 | z | < 1 𝑧 1 {\displaystyle{\displaystyle|z|<1}} 
abs(ln(1 + z)) <= - ln(1 -abs(z))

Abs[Log[1 + z]] <= - Log[1 -Abs[z]]
Failure Failure Successful [Tested: 1] Successful [Tested: 1]
4.5.E7 e - x / ( 1 - x ) < 1 - x superscript 𝑒 π‘₯ 1 π‘₯ 1 π‘₯ {\displaystyle{\displaystyle e^{-x/(1-x)}<1-x}}
e^{-x/(1-x)} < 1-x		 x < 1 π‘₯ 1 {\displaystyle{\displaystyle x<1}} 
exp(- x/(1 - x)) < 1 - x

Exp[- x/(1 - x)] < 1 - x
Skipped - no semantic math Failure - Successful [Tested: 1]
4.5.E7 1 - x < e - x 1 π‘₯ superscript 𝑒 π‘₯ {\displaystyle{\displaystyle 1-x<e^{-x}}}
1-x < e^{-x}		 x < 1 π‘₯ 1 {\displaystyle{\displaystyle x<1}} 
1 - x < exp(- x)

1 - x < Exp[- x]
Error Failure - Successful [Tested: 1]
4.5.E8 1 + x < e x 1 π‘₯ superscript 𝑒 π‘₯ {\displaystyle{\displaystyle 1+x<e^{x}}}
1+x < e^{x}		 - ∞ < x , x < ∞ formulae-sequence π‘₯ π‘₯ {\displaystyle{\displaystyle-\infty<x,x<\infty}} 
1 + x < exp(x)

1 + x < Exp[x]
Skipped - no semantic math Failure - Successful [Tested: 3]
4.5.E9 e x < 1 1 - x superscript 𝑒 π‘₯ 1 1 π‘₯ {\displaystyle{\displaystyle e^{x}<\frac{1}{1-x}}}
e^{x} < \frac{1}{1-x}		 x < 1 π‘₯ 1 {\displaystyle{\displaystyle x<1}} 
exp(x) < (1)/(1 - x)

Exp[x] < Divide[1,1 - x]
Skipped - no semantic math Failure - Successful [Tested: 1]
4.5.E10 x 1 + x < 1 - e - x π‘₯ 1 π‘₯ 1 superscript 𝑒 π‘₯ {\displaystyle{\displaystyle\frac{x}{1+x}<1-e^{-x}}}
\frac{x}{1+x} < 1-e^{-x}		 x > - 1 π‘₯ 1 {\displaystyle{\displaystyle x>-1}} 
(x)/(1 + x) < 1 - exp(- x)

Divide[x,1 + x] < 1 - Exp[- x]
Skipped - no semantic math Failure - Successful [Tested: 3]
4.5.E10 1 - e - x < x 1 superscript 𝑒 π‘₯ π‘₯ {\displaystyle{\displaystyle 1-e^{-x}<x}}
1-e^{-x} < x		 x > - 1 π‘₯ 1 {\displaystyle{\displaystyle x>-1}} 
1 - exp(- x) < x

1 - Exp[- x] < x
Error Failure - Successful [Tested: 3]
4.5.E11 x < e x - 1 π‘₯ superscript 𝑒 π‘₯ 1 {\displaystyle{\displaystyle x<e^{x}-1}}
x < e^{x}-1		 x < 1 π‘₯ 1 {\displaystyle{\displaystyle x<1}} 
x < exp(x)- 1

x < Exp[x]- 1
Skipped - no semantic math Failure - Successful [Tested: 1]
4.5.E11 e x - 1 < x 1 - x superscript 𝑒 π‘₯ 1 π‘₯ 1 π‘₯ {\displaystyle{\displaystyle e^{x}-1<\frac{x}{1-x}}}
e^{x}-1 < \frac{x}{1-x}		 x < 1 π‘₯ 1 {\displaystyle{\displaystyle x<1}} 
exp(x)- 1 < (x)/(1 - x)

Exp[x]- 1 < Divide[x,1 - x]
Error Failure - Successful [Tested: 1]
4.5.E12 e x / ( 1 + x ) < 1 + x superscript 𝑒 π‘₯ 1 π‘₯ 1 π‘₯ {\displaystyle{\displaystyle e^{x/(1+x)}<1+x}}
e^{x/(1+x)} < 1+x		 x > - 1 π‘₯ 1 {\displaystyle{\displaystyle x>-1}} 
exp(x/(1 + x)) < 1 + x

Exp[x/(1 + x)] < 1 + x
Skipped - no semantic math Failure - Successful [Tested: 3]
4.5.E13 e x ⁒ y / ( x + y ) < ( 1 + x y ) y superscript 𝑒 π‘₯ 𝑦 π‘₯ 𝑦 superscript 1 π‘₯ 𝑦 𝑦 {\displaystyle{\displaystyle e^{xy/(x+y)}<\left(1+\frac{x}{y}\right)^{y}}}
e^{xy/(x+y)} < \left(1+\frac{x}{y}\right)^{y}		 x > 0 , y > 0 formulae-sequence π‘₯ 0 𝑦 0 {\displaystyle{\displaystyle x>0,y>0}} 
exp(x*y/(x + y)) < (1 +(x)/(y))^(y)

Exp[x*y/(x + y)] < (1 +Divide[x,y])^(y)
Skipped - no semantic math Failure - Successful [Tested: 9]
4.5.E13 ( 1 + x y ) y < e x superscript 1 π‘₯ 𝑦 𝑦 superscript 𝑒 π‘₯ {\displaystyle{\displaystyle\left(1+\frac{x}{y}\right)^{y}<e^{x}}}
\left(1+\frac{x}{y}\right)^{y} < e^{x}		 x > 0 , y > 0 formulae-sequence π‘₯ 0 𝑦 0 {\displaystyle{\displaystyle x>0,y>0}} 
(1 +(x)/(y))^(y) < exp(x)

(1 +Divide[x,y])^(y) < Exp[x]
Error Failure - Successful [Tested: 9]
4.5.E14 e - x < 1 - 1 2 ⁒ x superscript 𝑒 π‘₯ 1 1 2 π‘₯ {\displaystyle{\displaystyle e^{-x}<1-\tfrac{1}{2}x}}
e^{-x} < 1-\tfrac{1}{2}x		 0 < x , x ≀ 1.5936 ⁒ … formulae-sequence 0 π‘₯ π‘₯ 1.5936 … {\displaystyle{\displaystyle 0<x,x\leq 1.5936\dots}} 
exp(- x) < 1 -(1)/(2)*x

Exp[- x] < 1 -Divide[1,2]*x
Skipped - no semantic math Failure - Successful [Tested: 2]
4.5.E15 1 4 ⁒ | z | < | e z - 1 | 1 4 𝑧 superscript 𝑒 𝑧 1 {\displaystyle{\displaystyle\tfrac{1}{4}|z|<|e^{z}-1|}}
\tfrac{1}{4}|z| < |e^{z}-1|		 0 < | z | , | z | < 1 formulae-sequence 0 𝑧 𝑧 1 {\displaystyle{\displaystyle 0<|z|,|z|<1}} 
(1)/(4)*abs(z) < abs(exp(z)- 1)

Divide[1,4]*Abs[z] < Abs[Exp[z]- 1]
Skipped - no semantic math Failure - Successful [Tested: 1]
4.5.E15 | e z - 1 | < 7 4 ⁒ | z | superscript 𝑒 𝑧 1 7 4 𝑧 {\displaystyle{\displaystyle|e^{z}-1|<\tfrac{7}{4}|z|}}
|e^{z}-1| < \tfrac{7}{4}|z|		 0 < | z | , | z | < 1 formulae-sequence 0 𝑧 𝑧 1 {\displaystyle{\displaystyle 0<|z|,|z|<1}} 
abs(exp(z)- 1) < (7)/(4)*abs(z)

Abs[Exp[z]- 1] < Divide[7,4]*Abs[z]
Error Failure - Successful [Tested: 1]
4.5.E16 | e z - 1 | ≀ e | z | - 1 superscript 𝑒 𝑧 1 superscript 𝑒 𝑧 1 {\displaystyle{\displaystyle|e^{z}-1|\leq e^{|z|}-1}}
|e^{z}-1| \leq e^{|z|}-1		 

abs(exp(z)- 1) <= exp(abs(z))- 1

Abs[Exp[z]- 1] <= Exp[Abs[z]]- 1
Skipped - no semantic math Failure - Successful [Tested: 1]
4.5.E16 e | z | - 1 ≀ | z | ⁒ e | z | superscript 𝑒 𝑧 1 𝑧 superscript 𝑒 𝑧 {\displaystyle{\displaystyle e^{|z|}-1\leq|z|e^{|z|}}}
e^{|z|}-1 \leq |z|e^{|z|}		 

exp(abs(z))- 1 <= abs(z)*exp(abs(z))

Exp[Abs[z]]- 1 <= Abs[z]*Exp[Abs[z]]
Error Failure - Successful [Tested: 1]
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