Gamma Function - 5.6 Inequalities

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
5.6.E1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 1 < (2\pi)^{-1/2}x^{(1/2)-x}e^{x}\EulerGamma@{x}}
1 < (2\pi)^{-1/2}x^{(1/2)-x}e^{x}\EulerGamma@{x}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{x} > 0} 
1 < (2*Pi)^(- 1/2)* (x)^((1/2)- x)* exp(x)*GAMMA(x)

1 < (2*Pi)^(- 1/2)* (x)^((1/2)- x)* Exp[x]*Gamma[x]
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
5.6.E1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle (2\pi)^{-1/2}x^{(1/2)-x}e^{x}\EulerGamma@{x} < e^{1/(12x)}}
(2\pi)^{-1/2}x^{(1/2)-x}e^{x}\EulerGamma@{x} < e^{1/(12x)}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{x} > 0} 
(2*Pi)^(- 1/2)* (x)^((1/2)- x)* exp(x)*GAMMA(x) < exp(1/(12*x))

(2*Pi)^(- 1/2)* (x)^((1/2)- x)* Exp[x]*Gamma[x] < Exp[1/(12*x)]
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
5.6.E2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{1}{\EulerGamma@{x}}+\frac{1}{\EulerGamma@{1/x}} \leq 2}
\frac{1}{\EulerGamma@{x}}+\frac{1}{\EulerGamma@{1/x}} \leq 2		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{x} > 0, \realpart@@{(1/x)} > 0} 
(1)/(GAMMA(x))+(1)/(GAMMA(1/x)) <= 2

Divide[1,Gamma[x]]+Divide[1,Gamma[1/x]] <= 2
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
5.6.E3 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{1}{(\EulerGamma@{x})^{2}}+\frac{1}{(\EulerGamma@{1/x})^{2}} \leq 2}
\frac{1}{(\EulerGamma@{x})^{2}}+\frac{1}{(\EulerGamma@{1/x})^{2}} \leq 2		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{x} > 0, \realpart@@{(1/x)} > 0} 
(1)/((GAMMA(x))^(2))+(1)/((GAMMA(1/x))^(2)) <= 2

Divide[1,(Gamma[x])^(2)]+Divide[1,(Gamma[1/x])^(2)] <= 2
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
5.6.E4 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{\EulerGamma@{x+1}}{\EulerGamma@{x+s}} < (x+1)^{1-s}}
\frac{\EulerGamma@{x+1}}{\EulerGamma@{x+s}} < (x+1)^{1-s}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 0 < s, s < 1, \realpart@@{(x+1)} > 0, \realpart@@{(x+s)} > 0} 
(GAMMA(x + 1))/(GAMMA(x + s)) < (x + 1)^(1 - s)

Divide[Gamma[x + 1],Gamma[x + s]] < (x + 1)^(1 - s)
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
5.6.E5 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \exp@{(1-s)\digamma@{x+s^{1/2}}} \leq \frac{\EulerGamma@{x+1}}{\EulerGamma@{x+s}}}
\exp@{(1-s)\digamma@{x+s^{1/2}}} \leq \frac{\EulerGamma@{x+1}}{\EulerGamma@{x+s}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 0 < s, s < 1, \realpart@@{(x+1)} > 0, \realpart@@{(x+s)} > 0} 
exp((1 - s)*Psi(x + (s)^(1/2))) <= (GAMMA(x + 1))/(GAMMA(x + s))

Exp[(1 - s)*PolyGamma[x + (s)^(1/2)]] <= Divide[Gamma[x + 1],Gamma[x + s]]
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
5.6.E5 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{\EulerGamma@{x+1}}{\EulerGamma@{x+s}} \leq \exp@{(1-s)\digamma@{x+\tfrac{1}{2}(s+1)}}}
\frac{\EulerGamma@{x+1}}{\EulerGamma@{x+s}} \leq \exp@{(1-s)\digamma@{x+\tfrac{1}{2}(s+1)}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 0 < s, s < 1, \realpart@@{(x+1)} > 0, \realpart@@{(x+s)} > 0} 
(GAMMA(x + 1))/(GAMMA(x + s)) <= exp((1 - s)*Psi(x +(1)/(2)*(s + 1)))

Divide[Gamma[x + 1],Gamma[x + s]] <= Exp[(1 - s)*PolyGamma[x +Divide[1,2]*(s + 1)]]
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
5.6.E6 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |\EulerGamma@{x+\iunit y}| \leq |\EulerGamma@{x}|}
|\EulerGamma@{x+\iunit y}| \leq |\EulerGamma@{x}|		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{(x+\iunit y)} > 0, \realpart@@{x} > 0} 
abs(GAMMA(x + I*y)) <= abs(GAMMA(x))

Abs[Gamma[x + I*y]] <= Abs[Gamma[x]]
Failure Failure Successful [Tested: 18] Successful [Tested: 18]
5.6.E7 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |\EulerGamma@{x+\iunit y}| \geq (\sech@{\pi y})^{1/2}\EulerGamma@{x}}
|\EulerGamma@{x+\iunit y}| \geq (\sech@{\pi y})^{1/2}\EulerGamma@{x}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle x \geq \tfrac{1}{2}, \realpart@@{(x+\iunit y)} > 0, \realpart@@{x} > 0} 
abs(GAMMA(x + I*y)) >= (sech(Pi*y))^(1/2)* GAMMA(x)

Abs[Gamma[x + I*y]] >= (Sech[Pi*y])^(1/2)* Gamma[x]
Failure Failure Successful [Tested: 18] Successful [Tested: 18]
5.6.E8 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \left|\frac{\EulerGamma@{z+a}}{\EulerGamma@{z+b}}\right| \leq \frac{1}{|z|^{b-a}}}
\left|\frac{\EulerGamma@{z+a}}{\EulerGamma@{z+b}}\right| \leq \frac{1}{|z|^{b-a}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{(z+a)} > 0, \realpart@@{(z+b)} > 0} 
abs((GAMMA(z + a))/(GAMMA(z + b))) <= (1)/((abs(z))^(b - a))

Abs[Divide[Gamma[z + a],Gamma[z + b]]] <= Divide[1,(Abs[z])^(b - a)]
Failure Failure
Failed [30 / 83]
Result: .5333333334 <= .1250000000
Test Values: {a = -1.5, b = 1.5, z = 2}


Result: 2.000000000 <= .5000000000
Test Values: {a = -1.5, b = -.5, z = 2}


Result: 1.333333334 <= .2500000000
Test Values: {a = -1.5, b = .5, z = 2}


Result: .2954089752 <= .8838834764e-1
Test Values: {a = -1.5, b = 2, z = 2}

... skip entries to safe data
Failed [35 / 95]
Result: False
Test Values: {Rule[a, -1.5], Rule[b, 1.5], Rule[z, 2]}


Result: False
Test Values: {Rule[a, -1.5], Rule[b, -0.5], Rule[z, 2]}

... skip entries to safe data
5.6.E9 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |\EulerGamma@{z}| \leq (2\pi)^{1/2}|z|^{x-(1/2)}e^{-\pi|y|/2}\exp@{\tfrac{1}{6}|z|^{-1}}}
|\EulerGamma@{z}| \leq (2\pi)^{1/2}|z|^{x-(1/2)}e^{-\pi|y|/2}\exp@{\tfrac{1}{6}|z|^{-1}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{z} > 0} 
abs(GAMMA(x + y*I)) <= (2*Pi)^(1/2)*(abs(x + y*I))^(x -(1/2))* exp(- Pi*abs(y)/2)*exp((1)/(6)*(abs(x + y*I))^(- 1))

Abs[Gamma[x + y*I]] <= (2*Pi)^(1/2)*(Abs[x + y*I])^(x -(1/2))* Exp[- Pi*Abs[y]/2]*Exp[Divide[1,6]*(Abs[x + y*I])^(- 1)]
Failure Failure Successful [Tested: 18] Successful [Tested: 18]
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