Incomplete Gamma and Related Functions - 8.10 Inequalities

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
8.10.E1 x 1 - a ⁢ e x ⁢ Γ ⁡ ( a , x ) ≤ 1 superscript 𝑥 1 𝑎 superscript 𝑒 𝑥 incomplete-Gamma 𝑎 𝑥 1 {\displaystyle{\displaystyle x^{1-a}e^{x}\Gamma\left(a,x\right)\leq 1}}
x^{1-a}e^{x}\incGamma@{a}{x} \leq 1		 x > 0 , 0 < a , a ≤ 1 formulae-sequence 𝑥 0 formulae-sequence 0 𝑎 𝑎 1 {\displaystyle{\displaystyle x>0,0<a,a\leq 1}} 
(x)^(1 - a)* exp(x)*GAMMA(a, x) <= 1

(x)^(1 - a)* Exp[x]*Gamma[a, x] <= 1
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
8.10.E2 γ ⁡ ( a , x ) ≥ x a - 1 a ⁢ ( 1 - e - x ) incomplete-gamma 𝑎 𝑥 superscript 𝑥 𝑎 1 𝑎 1 superscript 𝑒 𝑥 {\displaystyle{\displaystyle\gamma\left(a,x\right)\geq\frac{x^{a-1}}{a}(1-e^{-% x})}}
\incgamma@{a}{x} \geq \frac{x^{a-1}}{a}(1-e^{-x})		 x > 0 , 0 < a , a ≤ 1 , ℜ ⁡ a > 0 formulae-sequence 𝑥 0 formulae-sequence 0 𝑎 formulae-sequence 𝑎 1 𝑎 0 {\displaystyle{\displaystyle x>0,0<a,a\leq 1,\Re a>0}} 
GAMMA(a)-GAMMA(a, x) >= ((x)^(a - 1))/(a)*(1 - exp(- x))

Gamma[a, 0, x] >= Divide[(x)^(a - 1),a]*(1 - Exp[- x])
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
8.10.E3 x 1 - a ⁢ e x ⁢ Γ ⁡ ( a , x ) = 1 + a - 1 x ⁢ ϑ superscript 𝑥 1 𝑎 superscript 𝑒 𝑥 incomplete-Gamma 𝑎 𝑥 1 𝑎 1 𝑥 italic-ϑ {\displaystyle{\displaystyle x^{1-a}e^{x}\Gamma\left(a,x\right)=1+\frac{a-1}{x% }\vartheta}}
x^{1-a}e^{x}\incGamma@{a}{x} = 1+\frac{a-1}{x}\vartheta		 

(x)^(1 - a)* exp(x)*GAMMA(a, x) = 1 +(a - 1)/(x)*vartheta

(x)^(1 - a)* Exp[x]*Gamma[a, x] == 1 +Divide[a - 1,x]*\[CurlyTheta]
Failure Failure
Failed [180 / 180]
Result: .8735840245+.8333333335*I
Test Values: {a = -1.5, x = 1.5, vartheta = 1/2*3^(1/2)+1/2*I}


Result: -1.403124983+1.443375674*I
Test Values: {a = -1.5, x = 1.5, vartheta = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [180 / 180]
Result: Complex[0.8735840235649492, 0.8333333333333331]
Test Values: {Rule[a, -1.5], Rule[x, 1.5], Rule[ϑ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-1.4031249827424481, 1.4433756729740643]
Test Values: {Rule[a, -1.5], Rule[x, 1.5], Rule[ϑ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
8.10.E4 0 < ϑ 0 italic-ϑ {\displaystyle{\displaystyle 0<\vartheta}}
0 < \vartheta		 x > 0 , a ≤ 2 formulae-sequence 𝑥 0 𝑎 2 {\displaystyle{\displaystyle x>0,a\leq 2}} 
0 < vartheta
 
0 < \[CurlyTheta]
 Skipped - no semantic math Skipped - no semantic math - -
8.10.E5 A n < x 1 - a ⁢ e x ⁢ Γ ⁡ ( a , x ) subscript 𝐴 𝑛 superscript 𝑥 1 𝑎 superscript 𝑒 𝑥 incomplete-Gamma 𝑎 𝑥 {\displaystyle{\displaystyle A_{n}<x^{1-a}e^{x}\Gamma\left(a,x\right)}}
A_{n} < x^{1-a}e^{x}\incGamma@{a}{x}		 x > 0 , a < 1 formulae-sequence 𝑥 0 𝑎 1 {\displaystyle{\displaystyle x>0,a<1}} 
A[n](<)*(x)^(1 - a)* exp(x)*GAMMA(a, x)

Subscript[A, n][<]*(x)^(1 - a)* Exp[x]*Gamma[a, x]
Failure Failure
Failed [75 / 300]
Result: 1.5 < .4302083505
Test Values: {a = -1.5, x = 1.5, A[n] = 1.5, n = 1}


Result: 1.5 < .4302083505
Test Values: {a = -1.5, x = 1.5, A[n] = 1.5, n = 2}

... skip entries to safe data
Failed [195 / 300]
Result: Less[Complex[0.8660254037844387, 0.49999999999999994], 0.43020835059088497]
Test Values: {Rule[a, -1.5], Rule[n, 1], Rule[x, 1.5], Rule[Subscript[A, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Less[Complex[0.8660254037844387, 0.49999999999999994], 0.43020835059088497]
Test Values: {Rule[a, -1.5], Rule[n, 2], Rule[x, 1.5], Rule[Subscript[A, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
8.10.E5 x 1 - a ⁢ e x ⁢ Γ ⁡ ( a , x ) < B n superscript 𝑥 1 𝑎 superscript 𝑒 𝑥 incomplete-Gamma 𝑎 𝑥 subscript 𝐵 𝑛 {\displaystyle{\displaystyle x^{1-a}e^{x}\Gamma\left(a,x\right)<B_{n}}}
x^{1-a}e^{x}\incGamma@{a}{x} < B_{n}		 x > 0 , a < 1 formulae-sequence 𝑥 0 𝑎 1 {\displaystyle{\displaystyle x>0,a<1}} 
(x)^(1 - a)* exp(x)*GAMMA(a, x) < B[n]

(x)^(1 - a)* Exp[x]*Gamma[a, x] < Subscript[B, n]
Failure Failure
Failed [105 / 300]
Result: .4302083505 < -1.5
Test Values: {a = -1.5, x = 1.5, B[n] = -1.5, n = 1}


Result: .4302083505 < -1.5
Test Values: {a = -1.5, x = 1.5, B[n] = -1.5, n = 2}

... skip entries to safe data
Failed [225 / 300]
Result: Less[0.43020835059088497, Complex[0.8660254037844387, 0.49999999999999994]]
Test Values: {Rule[a, -1.5], Rule[n, 1], Rule[x, 1.5], Rule[Subscript[B, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Less[0.43020835059088497, Complex[0.8660254037844387, 0.49999999999999994]]
Test Values: {Rule[a, -1.5], Rule[n, 2], Rule[x, 1.5], Rule[Subscript[B, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
8.10#Ex1 A 1 = x x + 1 - a subscript 𝐴 1 𝑥 𝑥 1 𝑎 {\displaystyle{\displaystyle A_{1}=\frac{x}{x+1-a}}}
A_{1} = \frac{x}{x+1-a}		 

A[1] = (x)/(x + 1 - a)
 
Subscript[A, 1] == Divide[x,x + 1 - a]
 Skipped - no semantic math Skipped - no semantic math - -
8.10#Ex2 B 1 = x + 1 x + 2 - a subscript 𝐵 1 𝑥 1 𝑥 2 𝑎 {\displaystyle{\displaystyle B_{1}=\frac{x+1}{x+2-a}}}
B_{1} = \frac{x+1}{x+2-a}		 

B[1] = (x + 1)/(x + 2 - a)
 
Subscript[B, 1] == Divide[x + 1,x + 2 - a]
 Skipped - no semantic math Skipped - no semantic math - -
8.10#Ex3 A 2 = x ⁢ ( x + 3 - a ) x 2 + 2 ⁢ ( 2 - a ) ⁢ x + ( 1 - a ) ⁢ ( 2 - a ) subscript 𝐴 2 𝑥 𝑥 3 𝑎 superscript 𝑥 2 2 2 𝑎 𝑥 1 𝑎 2 𝑎 {\displaystyle{\displaystyle A_{2}=\frac{x(x+3-a)}{x^{2}+2(2-a)x+(1-a)(2-a)}}}
A_{2} = \frac{x(x+3-a)}{x^{2}+2(2-a)x+(1-a)(2-a)}		 

A[2] = (x*(x + 3 - a))/((x)^(2)+ 2*(2 - a)*x +(1 - a)*(2 - a))
 
Subscript[A, 2] == Divide[x*(x + 3 - a),(x)^(2)+ 2*(2 - a)*x +(1 - a)*(2 - a)]
 Skipped - no semantic math Skipped - no semantic math - -
8.10#Ex4 B 2 = x 2 + ( 5 - a ) ⁢ x + 2 x 2 + 2 ⁢ ( 3 - a ) ⁢ x + ( 2 - a ) ⁢ ( 3 - a ) subscript 𝐵 2 superscript 𝑥 2 5 𝑎 𝑥 2 superscript 𝑥 2 2 3 𝑎 𝑥 2 𝑎 3 𝑎 {\displaystyle{\displaystyle B_{2}=\frac{x^{2}+(5-a)x+2}{x^{2}+2(3-a)x+(2-a)(3% -a)}}}
B_{2} = \frac{x^{2}+(5-a)x+2}{x^{2}+2(3-a)x+(2-a)(3-a)}		 

B[2] = ((x)^(2)+(5 - a)*x + 2)/((x)^(2)+ 2*(3 - a)*x +(2 - a)*(3 - a))
 
Subscript[B, 2] == Divide[(x)^(2)+(5 - a)*x + 2,(x)^(2)+ 2*(3 - a)*x +(2 - a)*(3 - a)]
 Skipped - no semantic math Skipped - no semantic math - -
8.10.E7 I = ∫ 0 x t a - 1 ⁢ e t ⁢ d t 𝐼 superscript subscript 0 𝑥 superscript 𝑡 𝑎 1 superscript 𝑒 𝑡 𝑡 {\displaystyle{\displaystyle I=\int_{0}^{x}t^{a-1}e^{t}\mathrm{d}t}}
I = \int_{0}^{x}t^{a-1}e^{t}\diff{t}		 ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}} 
I = int((t)^(a - 1)* exp(t), t = 0..x)

I == Integrate[(t)^(a - 1)* Exp[t], {t, 0, x}, GenerateConditions->None]
Failure Failure
Failed [90 / 90]
Result: -2.374226751+.5000000000*I
Test Values: {I = 1/2*3^(1/2)+1/2*I, a = 1.5, x = 1.5}


Result: .5451660792+.5000000000*I
Test Values: {I = 1/2*3^(1/2)+1/2*I, a = 1.5, x = .5}

... skip entries to safe data
Failed [27 / 27]
Result: Complex[-2.240252154794788, -4.0113152157384396*^-17]
Test Values: {Rule[Complex[0, 1], 1], Rule[a, 1.5], Rule[x, 1.5]}


Result: Complex[-1.240252154794788, -4.0113152157384396*^-17]
Test Values: {Rule[Complex[0, 1], 2], Rule[a, 1.5], Rule[x, 1.5]}

... skip entries to safe data
8.10.E7 ∫ 0 x t a - 1 ⁢ e t ⁢ d t = Γ ⁡ ( a ) ⁢ x a ⁢ γ * ⁡ ( a , - x ) superscript subscript 0 𝑥 superscript 𝑡 𝑎 1 superscript 𝑒 𝑡 𝑡 Euler-Gamma 𝑎 superscript 𝑥 𝑎 incomplete-gamma-star 𝑎 𝑥 {\displaystyle{\displaystyle\int_{0}^{x}t^{a-1}e^{t}\mathrm{d}t=\Gamma\left(a% \right)x^{a}\gamma^{*}\left(a,-x\right)}}
\int_{0}^{x}t^{a-1}e^{t}\diff{t} = \EulerGamma@{a}x^{a}\scincgamma@{a}{-x}		 ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}} 
int((t)^(a - 1)* exp(t), t = 0..x) = GAMMA(a)*(x)^(a)* (- x)^(-(a))*(GAMMA(a)-GAMMA(a, - x))/GAMMA(a)

Error
Failure Missing Macro Error Successful [Tested: 9] Skip - symbolical successful subtest
8.10.E8 ( a + 1 ) ⁢ ( a + 2 ) - x ( a + 1 ) ⁢ ( a + 2 + x ) < a ⁢ x - a ⁢ e - x ⁢ I 𝑎 1 𝑎 2 𝑥 𝑎 1 𝑎 2 𝑥 𝑎 superscript 𝑥 𝑎 superscript 𝑒 𝑥 𝐼 {\displaystyle{\displaystyle\frac{(a+1)(a+2)-x}{(a+1)(a+2+x)}<ax^{-a}e^{-x}I}}
\frac{(a+1)(a+2)-x}{(a+1)(a+2+x)} < ax^{-a}e^{-x}I		 x > 0 , a ≥ 0 formulae-sequence 𝑥 0 𝑎 0 {\displaystyle{\displaystyle x>0,a\geq 0}} 
((a + 1)*(a + 2)- x)/((a + 1)*(a + 2 + x)) < a*(x)^(- a)* exp(- x)*I
 
Divide[(a + 1)*(a + 2)- x,(a + 1)*(a + 2 + x)] < a*(x)^(- a)* Exp[- x]*I
 Skipped - no semantic math Skipped - no semantic math - -
8.10#Ex5 c a = ( Γ ⁡ ( 1 + a ) ) 1 / ( a - 1 ) subscript 𝑐 𝑎 superscript Euler-Gamma 1 𝑎 1 𝑎 1 {\displaystyle{\displaystyle c_{a}=(\Gamma\left(1+a\right))^{1/(a-1)}}}
c_{a} = (\EulerGamma@{1+a})^{1/(a-1)}		 ℜ ⁡ ( 1 + a ) > 0 1 𝑎 0 {\displaystyle{\displaystyle\Re(1+a)>0}} 
c[a] = (GAMMA(1 + a))^(1/(a - 1))

Subscript[c, a] == (Gamma[1 + a])^(1/(a - 1))
Failure Failure
Failed [39 / 40]
Result: -.9011204630+.5000000000*I
Test Values: {a = 1.5, c[a] = 1/2*3^(1/2)+1/2*I}


Result: -2.267145867+.8660254040*I
Test Values: {a = 1.5, c[a] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [39 / 40]
Result: Complex[-0.9011204638598199, 0.49999999999999994]
Test Values: {Rule[a, 1.5], Rule[Subscript[c, a], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-2.2671458676442584, 0.8660254037844387]
Test Values: {Rule[a, 1.5], Rule[Subscript[c, a], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
8.10#Ex6 d a = ( Γ ⁡ ( 1 + a ) ) - 1 / a subscript 𝑑 𝑎 superscript Euler-Gamma 1 𝑎 1 𝑎 {\displaystyle{\displaystyle d_{a}=(\Gamma\left(1+a\right))^{-1/a}}}
d_{a} = (\EulerGamma@{1+a})^{-1/a}		 ℜ ⁡ ( 1 + a ) > 0 1 𝑎 0 {\displaystyle{\displaystyle\Re(1+a)>0}} 
d[a] = (GAMMA(1 + a))^(- 1/a)

Subscript[d, a] == (Gamma[1 + a])^(- 1/a)
Failure Failure
Failed [40 / 40]
Result: .388914161e-1+.5000000000*I
Test Values: {a = 1.5, d[a] = 1/2*3^(1/2)+1/2*I}


Result: -1.327133988+.8660254040*I
Test Values: {a = 1.5, d[a] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [40 / 40]
Result: Complex[0.038891415918572037, 0.49999999999999994]
Test Values: {Rule[a, 1.5], Rule[Subscript[d, a], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-1.3271339878658663, 0.8660254037844387]
Test Values: {Rule[a, 1.5], Rule[Subscript[d, a], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
8.10.E10 x 2 ⁢ a ⁢ ( ( 1 + 2 x ) a - 1 ) < x 1 - a ⁢ e x ⁢ Γ ⁡ ( a , x ) 𝑥 2 𝑎 superscript 1 2 𝑥 𝑎 1 superscript 𝑥 1 𝑎 superscript 𝑒 𝑥 incomplete-Gamma 𝑎 𝑥 {\displaystyle{\displaystyle\frac{x}{2a}\left(\left(1+\frac{2}{x}\right)^{a}-1% \right)<x^{1-a}e^{x}\Gamma\left(a,x\right)}}
\frac{x}{2a}\left(\left(1+\frac{2}{x}\right)^{a}-1\right) < x^{1-a}e^{x}\incGamma@{a}{x}		 x ≥ 0 , 0 < a , a < 1 formulae-sequence 𝑥 0 formulae-sequence 0 𝑎 𝑎 1 {\displaystyle{\displaystyle x\geq 0,0<a,a<1}} 
(x)/(2*a)*((1 +(2)/(x))^(a)- 1) < (x)^(1 - a)* exp(x)*GAMMA(a, x)

Divide[x,2*a]*((1 +Divide[2,x])^(a)- 1) < (x)^(1 - a)* Exp[x]*Gamma[a, x]
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
8.10.E10 x 1 - a ⁢ e x ⁢ Γ ⁡ ( a , x ) ≤ x a ⁢ c a ⁢ ( ( 1 + c a x ) a - 1 ) superscript 𝑥 1 𝑎 superscript 𝑒 𝑥 incomplete-Gamma 𝑎 𝑥 𝑥 𝑎 subscript 𝑐 𝑎 superscript 1 subscript 𝑐 𝑎 𝑥 𝑎 1 {\displaystyle{\displaystyle x^{1-a}e^{x}\Gamma\left(a,x\right)\leq\frac{x}{ac% _{a}}\left(\left(1+\frac{c_{a}}{x}\right)^{a}-1\right)}}
x^{1-a}e^{x}\incGamma@{a}{x} \leq \frac{x}{ac_{a}}\left(\left(1+\frac{c_{a}}{x}\right)^{a}-1\right)		 x ≥ 0 , 0 < a , a < 1 formulae-sequence 𝑥 0 formulae-sequence 0 𝑎 𝑎 1 {\displaystyle{\displaystyle x\geq 0,0<a,a<1}} 
(x)^(1 - a)* exp(x)*GAMMA(a, x) <= (x)/(a*c[a])*((1 +(c[a])/(x))^(a)- 1)

(x)^(1 - a)* Exp[x]*Gamma[a, x] <= Divide[x,a*Subscript[c, a]]*((1 +Divide[Subscript[c, a],x])^(a)- 1)
Failure Failure
Failed [3 / 30]
Result: .8100694499 <= .7912878480
Test Values: {a = .5, x = 1.5, c[a] = 2}


Result: .6556795425 <= .6180339885
Test Values: {a = .5, x = .5, c[a] = 2}

... skip entries to safe data
Failed [18 / 30]
Result: LessEqual[0.8100694501969615, Complex[0.8808481919138387, -0.05137441674828974]]
Test Values: {Rule[a, 0.5], Rule[x, 1.5], Rule[Subscript[c, a], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: LessEqual[0.8100694501969615, Complex[1.0324243733930456, -0.1801654927820326]]
Test Values: {Rule[a, 0.5], Rule[x, 1.5], Rule[Subscript[c, a], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
8.10.E11 ( 1 - e - α a ⁢ x ) a ≤ P ⁡ ( a , x ) superscript 1 superscript 𝑒 subscript 𝛼 𝑎 𝑥 𝑎 incomplete-gamma-P 𝑎 𝑥 {\displaystyle{\displaystyle(1-e^{-\alpha_{a}x})^{a}\leq P\left(a,x\right)}}
(1-e^{-\alpha_{a}x})^{a} \leq \normincGammaP@{a}{x}		 x ≥ 0 , a > 0 formulae-sequence 𝑥 0 𝑎 0 {\displaystyle{\displaystyle x\geq 0,a>0}} 
(1 - exp(- alpha[a]*x))^(a) <= (GAMMA(a)-GAMMA(a, x))/GAMMA(a)

(1 - Exp[- Subscript[\[Alpha], a]*x])^(a) <= GammaRegularized[a, 0, x]
Failure Failure
Failed [78 / 270]
Result: .8461432717 <= .6083748237
Test Values: {a = 1.5, alpha = 1.5, x = 1.5, alpha[a] = 1.5}


Result: .9262567903 <= .6083748237
Test Values: {a = 1.5, alpha = 1.5, x = 1.5, alpha[a] = 2}

... skip entries to safe data
Failed [240 / 270]
Result: LessEqual[Complex[0.7016331692747775, 0.2500919864059583], 0.608374823728911]
Test Values: {Rule[a, 1.5], Rule[x, 1.5], Rule[α, 1.5], Rule[Subscript[α, a], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: LessEqual[Complex[-1.369574242346028, 2.679891945423719], 0.608374823728911]
Test Values: {Rule[a, 1.5], Rule[x, 1.5], Rule[α, 1.5], Rule[Subscript[α, a], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
8.10.E11 P ⁡ ( a , x ) ≤ ( 1 - e - β a ⁢ x ) a incomplete-gamma-P 𝑎 𝑥 superscript 1 superscript 𝑒 subscript 𝛽 𝑎 𝑥 𝑎 {\displaystyle{\displaystyle P\left(a,x\right)\leq(1-e^{-\beta_{a}x})^{a}}}
\normincGammaP@{a}{x} \leq (1-e^{-\beta_{a}x})^{a}		 x ≥ 0 , a > 0 formulae-sequence 𝑥 0 𝑎 0 {\displaystyle{\displaystyle x\geq 0,a>0}} 
(GAMMA(a)-GAMMA(a, x))/GAMMA(a) <= (1 - exp(- beta[a]*x))^(a)

GammaRegularized[a, 0, x] <= (1 - Exp[- Subscript[\[Beta], a]*x])^(a)
Failure Failure
Failed [30 / 270]
Result: .6083748237 <= .3832643966
Test Values: {a = 1.5, beta = 1.5, x = 1.5, beta[a] = .5}


Result: .1987480431 <= .1040340193
Test Values: {a = 1.5, beta = 1.5, x = .5, beta[a] = .5}

... skip entries to safe data
Failed [192 / 270]
Result: LessEqual[0.608374823728911, Complex[0.7016331692747775, 0.2500919864059583]]
Test Values: {Rule[a, 1.5], Rule[x, 1.5], Rule[β, 1.5], Rule[Subscript[β, a], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: LessEqual[0.608374823728911, Complex[-1.369574242346028, 2.679891945423719]]
Test Values: {Rule[a, 1.5], Rule[x, 1.5], Rule[β, 1.5], Rule[Subscript[β, a], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
8.10.E13 Γ ⁡ ( n , n ) Γ ⁡ ( n ) < 1 2 incomplete-Gamma 𝑛 𝑛 Euler-Gamma 𝑛 1 2 {\displaystyle{\displaystyle\frac{\Gamma\left(n,n\right)}{\Gamma\left(n\right)% }<\frac{1}{2}}}
\frac{\incGamma@{n}{n}}{\EulerGamma@{n}} < \frac{1}{2}		 ℜ ⁡ n > 0 𝑛 0 {\displaystyle{\displaystyle\Re n>0}} 
(GAMMA(n, n))/(GAMMA(n)) < (1)/(2)

Divide[Gamma[n, n],Gamma[n]] < Divide[1,2]
Failure Failure Successful [Tested: 3] Successful [Tested: 1]
8.10.E13 1 2 < Γ ⁡ ( n , n - 1 ) Γ ⁡ ( n ) 1 2 incomplete-Gamma 𝑛 𝑛 1 Euler-Gamma 𝑛 {\displaystyle{\displaystyle\frac{1}{2}<\frac{\Gamma\left(n,n-1\right)}{\Gamma% \left(n\right)}}}
\frac{1}{2} < \frac{\incGamma@{n}{n-1}}{\EulerGamma@{n}}		 ℜ ⁡ n > 0 𝑛 0 {\displaystyle{\displaystyle\Re n>0}} 
(1)/(2) < (GAMMA(n, n - 1))/(GAMMA(n))

Divide[1,2] < Divide[Gamma[n, n - 1],Gamma[n]]
Failure Failure Successful [Tested: 3] Successful [Tested: 1]
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